จำนวนจริง

94 ⌫ ⌫  ⌦
 ⌫     ⌫  

˜ª´ °¥nµŠš3ɸ ‹Š®µÁŽ˜‡Îµ˜°…°Š°­¤„µ¦˜n°Åžœ¸Ê

2x 3 d 9 ¨³ 2x 3 d 9

ª·›¸šµÎ ‹µ„ 2x 3 d 9

‹³Å—o 9 d 2x 3 d 9

9 3 d 2x 3 3 d 9 3

6 d 2x d 12

3dx d6

ÁŽ˜‡Îµ˜°…°Š°­¤„µ¦‡°º > 3,6@

­nªœ 2x 3 d 9 Ä®oœ„´ Á¦¥¸ œ¨°Š®µ‡Îµ˜°Á°Š

˜´ª°¥µn Šš4ɸ ‹Š®µÁŽ˜‡µÎ ˜°…°Š°­¤„µ¦˜n°Åžœ¸Ê

1. x 3 ! 2x 9

2. 2x 5 d 1

3x 1

1. ‹µ„ x 3 ! 2x 9

‹³Å—o x 3 2x 9 ®¦º° x 3 ! 2x 9

x 3 2x 9 ®¦º° 3x ! 12

6 x ®¦°º x ! 4

œÉ´œ‡°º x ! 6

ÁŽ˜‡µÎ ˜°…°Š°­¤„µ¦‡°º 6,f

2. ‹µ„ 2x 5 d 1
3x 1

‹³Å—o 2x 5 d 1

3x 1

2x 5 d 3x 1

2x 5 2 d 3x 12

2x 5 2 3x 1 2 d 0
2x 5 3x 1 2x 5 3x 1 d 0

5x 4 x 6 d 0
5x 4 x 6 t 0

+ -6 - 4/5 +

⌦ 95
⌦

96 ⌫ ⌫  ⌦
 ⌫     ⌫  

5. ®¨Šn „µ¦Á¦¸¥œ¦¼o
5.1 Á°„­µ¦ „f ®´—š¸É 12
5.2 ®œŠ´ ­º°Á¦¥¸ œ­µ¦³„µ¦Á¦¥¸ œ¦¼oÁ¡É·¤Á˜·¤ ‡–˜· «µ­˜¦r Á¨¤n 1 ´Êœ¤›´ ¥¤«„¹ ¬µžešÉ¸ 4

6. „¦³ªœ„µ¦ª—´ ¨³ž¦³Á¤œ· Ÿ¨ „µ¦ž¦³Á¤œ· Ÿ¨
1. œ´„Á¦¸¥œ˜°‡Îµ™µ¤Å—™o ¼„˜°o ŠÁžœ} ­nªœ¤µ„
„µ¦ª—´ Ÿ¨ 2. œ„´ Á¦¸¥œ­œÄ‹Â¨³˜´ÊŠÄ‹Á¦¥¸ œ
1. ­´ŠÁ„˜‹µ„„µ¦˜°‡Îµ™µ¤ 3. œ„´ Á¦¥¸ œšµÎ ŗo™„¼ ˜o°Šž¦³¤µ– 80 %
2. ­´ŠÁ„˜‹µ„‡ªµ¤­œÄ‹ 4. œ´„Á¦¥¸ œšÎµÅ—™o ¼„˜o°Šž¦³¤µ– 85 %
3. šµÎ Á°„­µ¦ f„®´—šÉ¸ 12
4. šµÎ ˚¥r f„®´— 2.7 Ĝ®œŠ´ ­º° 5. œ„´ Á¦¸¥œÂ­—Šª›· ¸šµÎ ŗo™¼„˜°o Šž¦³¤µ– 80 %

Á¦¥¸ œ­µ¦³„µ¦Á¦¸¥œ¦oÁ¼ ¡É·¤Á˜·¤
‡–˜· «µ­˜¦r
5. š—­° ­—Šª·›š¸ ε

7. œ´ š„¹ ®¨Š´ ­°œ
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8. „·‹„¦¦¤„µ¦Á­œ°Âœ³
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⌦ 97
⌦

Á°„­µ¦ „f ®—´ š¸É 12

ºÉ° …………………………………………….œ´Ê ………………………Á¨…šÉ¸ ………………………..
‹ŠÂ„°o ­¤„µ¦˜°n Şœ¸Ê
1. x 2 ! 5

2. x 2 5

3. 3 4x d 0

4. 2x 3 4x

5. 5x 4 d 2 3x

6. 12x 7 d 5x 2

7. 3 5x 2 3x

8. x 1 d x 2

9. x 4 2

x 1

10. 2x 1 t 3

x 1

11. x 1 x 1 1

x 1

12. 1

x

13. x 1

x 5

98 ⌫ ⌫  ⌦
 ⌫     ⌫  

Ÿœ„µ¦‹—´ „µ¦Á¦¥¸ œ¦¼šo ɸ 13

Á¦°Éº Š ­´‹¡‹œr…°Š‡ªµ¤¦·¼¦–r œ´Ê ¤´›¥¤«„¹ ¬µžše ¸É 4
ª·µ ‡–˜· «µ­˜¦r Áª¨µ 2 Éª´ äŠ

***********************************************************************************

Ÿ¨„µ¦Á¦¸¥œ¦šo¼ ‡É¸ µ—®ª´Š
°„­¤˜´ ‡· ªµ¤¦·¦¼ –rŗo

1. ‹—» ž¦³­Š‡„r µ¦Á¦¥¸ œ¦o¼ œ´„Á¦¸¥œ­µ¤µ¦™
°„œ·¥µ¤…°Š…°Á…˜œÂ¨³¦³…» °Á…˜œ…°Š­´ÁŽ˜…°Š R š„¸É µÎ ®œ—Ä®Åo —o
®µ…°Á…˜œ‡µn œ°o ¥­»—…°ŠÁŽ˜š¤É¸ …¸ °Á…˜œÅ—o
°„œ¥· µ¤…°Š…°Á…˜¨nµŠÂ¨³¦³…» °Á…˜¨nµŠ…°Š­´ÁŽ˜…°Š R š„ɸ µÎ ®œ—Ä®Åo —o
®µ…°Á…˜¨nµŠ‡µn ¤µ„­—» …°ŠÁŽ˜šÉ¸¤…¸ °Á…˜¨nµŠÅ—o

2. œª‡ªµ¤‡—· ®¨´„
­´‹¡‹œr…°Š‡ªµ¤¦·¼¦–rÁž}œ­¤´˜·ž¦³„µ¦­»—šoµ¥…°Š¦³‹Îµœªœ‹¦·Š Ž¹ÉŠ¦³‹Îµœªœ°Éºœ

Ťn¤¸ ¨³¤¸Éº°Á¦¸¥„°¸„°¥nµŠ®œ¹ÉŠªnµ ­´‹¡‹œr„µ¦¤¸‡nµ…°Á…˜œœo°¥­»— ­´‹¡‹œr…°Š‡ªµ¤¦·¼¦–r‹³Äo
Áž}œ­nªœ®œŠ¹É …°Š¦µ„“µœ‡–·˜«µ­˜¦rœÊ´ ­¼ŠµŠ­µ…µ

3. Áœº°Ê ®µ­µ¦³

šœ¥· µ¤ 1 ™oµ S Ážœ} ­´ÁŽ˜…°Š R S ‹³¤…¸ °Á…˜œ„˜È °n Á¤É°º ¤‹¸ µÎ œªœ‹¦·Š a ŽŠÉ¹ x d a
­Îµ®¦´ ‹µÎ œªœ‹¦Š· x š»„˜´ªÄœ S Á¦¥¸ „‹Îµœªœ‹¦·Š a œÊª¸ nµ…°Á…˜œ (upper bound) …°Š S

­´‹¡‹œ…r °Š‡ªµ¤¦·¼¦–r
™µo S  R , S z I ¨³ S ¤¸…°Á…˜œÂ¨oª S ‹³¤¸…°Á…˜œ‡µn œo°¥­—»

šœ¥· µ¤ 2 ™oµ S Ážœ} ­´ÁŽ˜…°Š R ‹Îµœªœ‹¦Š· a ‹³Áž}œ…°Á…˜œ‡µn œ°o ¥­»— (Least
upper bound) ®¦°º supremum …°Š S „˜È n°Á¤Éº°

1. a Áž}œ…°Á…˜œ…°Š S
¨³ 2. ™µo b Áž}œ…°Á…˜œ‡nµœ°o ¥šÉ­¸ —» …°Š S ‹³Å—ªo nµ bd a Á…¥¸ œÂšœ…°Á…˜œ‡µn
œ°o ¥­»—…°Š S —ªo ¥ sup S (®¦º° lub S)

⌦ 99
⌦

Ánœ ™oµ S = (1,4 ) ‹³Å—o sup S = 4

šœ¥· µ¤ 3 ™oµ S Ážœ} ­´ÁŽ˜…°Š R S ‹³¤…¸ °Á…˜¨nµŠ „˜È °n Á¤°ºÉ ¤¸‹µÎ œªœ‹¦·Š a Ž¹ÉŠ
a d x ­µÎ ®¦´ ‹µÎ œªœ‹¦Š· x š»„˜´ªÄœ S Á¦¸¥„‹Îµœªœ‹¦Š· a ªµn …°Á…˜¨nµŠ (lower bound) …°Š S

šœ¥· µ¤ 4 ™oµ S Ážœ} ­´ÁŽ˜ R ‹Îµœªœ‹¦Š· a ‹³Ážœ} …°Á…˜¨nµŠ‡nµ¤µ„­»— (greatest lower
bound ®¦°º infimum) …°Š S „˜È n°Á¤°Éº

1. a Áž}œ…°Á…˜¨µn Š…°Š S
¨³ 2. ™oµ b Ážœ} …°Á…˜¨nµŠ‡nµ¤µ„š¸É­»—…°Š S ‹³Å—ªo µn a d b

Á…¥¸ œÂšœ…°Á…˜¨µn Š‡nµ¤µ„­—» …°Š S —oª¥ inf S (®¦°º glb S) Ánœ
™µo S = (6,9) ‹³Å—o inf S = 6

4. „¦³ªœ„µ¦‹—´ „µ¦Á¦¸¥œ¦o¼
4.1 ššªœ­¤˜´ ·…°Š‹µÎ œªœ‹¦Š· šÉÁ¸ ¦¥¸ œŸµn œ¤µÂ¨ªo ×¥‡¦¼™µ¤Ä®oœ´„Á¦¸¥œ˜°š¨¸ ³‡œ
4.2 Ä®oœ„´ Á¦¸¥œšµÎ Á°„­µ¦Âœ³ÂœªšµŠš¸É 13 čoÁª¨µ 10 œµš¸
4.3 ‡¦¼ÁŒ¨¥‡Îµ˜°Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 13 ¨³°›·µ¥Ä®oœ´„Á¦¸¥œš¦µªnµ …o°Ä—šÉ¸°„Å—oªnµ

¤‹¸ µÎ œªœ‹¦·ŠÄ—šÉ¸Å¤nœo°¥„ªnµ‹µÎ œªœÄ—Ç ĜÁŽ˜œ´œÊ Á¦µÁ¦¥¸ „ÁŽ˜œœ´Ê ªnµÁžœ} ÁŽ˜š¤É¸ …¸ °Á…˜œ
4.4 ‡¦¼°›·µ¥ šœ·¥µ¤ 1- 2 ­´‹¡‹œr…°Š‡ªµ¤¦·¼¦–r ¡¦o°¤šÊ´Š¥„˜´ª°¥nµŠ °›·µ¥Ã—¥Äo

ª›· „¸ µ¦™µ¤˜°—Š´ ˜°n ޜʸ
˜´ª°¥µn Š Ä®oœ´„Á¦¸¥œnª¥„´œ®µ…°Á…˜œ Ĝ„¦–¸šÉ¸¤¸…°Á…˜œ ‹³¤¸…°Á…˜œ‡nµœo°¥­»—

®¦º°Å¤n (ĜªŠÁ¨ÈÁž}œ‡Îµ˜°)

1) 2,5@ ( ¤¸…°Á…˜œÁžœ} ‹Îµœªœ‹¦Š· š¸¤É µ„„ªµn ®¦°º Ášµn „´ 5 ¨³…°Á…˜œ‡µn œ°o ¥

­—» ‡º° 5 )

2) f, 5 ( ¤…¸ °Á…˜œÁž}œ‹Îµœªœ‹¦Š· šÉ¸¤µ„„ªµn ®¦º°Ášnµ„´ -5 ¨³…°Á…˜œ‡nµœo°¥

­—» ‡º° –5)

3) > 10,f (Ť¤n ‡¸ nµ…°Á…˜œ )

4) I ( ¤¸‡nµ…°Á…˜œÁž}œ‹µÎ œªœ‹¦·Šš»„‹µÎ œªœ ˜nŤ¤n ¸ …°Á…˜œ‡nµœ°o ¥­—» )

5) ­®x | x 2 1 ,n I,n t 0¾½
¯ 2n ¿

(2 1 2 š»„Ç‹µÎ œªœÁ˜¤È n t 0 —´Šœ´ÊœÁŽ˜œÊ¸¤…¸ °Á…˜œ ¨³‡nµ…°Á…˜œœ°o ¥­—» ‡º° 2 )
2n

100 ⌫ ⌫  ⌦
 ⌫     ⌫  

6) ^1,2,3,4`‰ ^ 5,S `
(­¤µ·„š„» ˜ª´ …°Š ^1,2,3,4`‰^ 5,S `¤‡¸ nµ…°Á…˜œÂ¨³‡µn …°Á…˜œœ°o ¥­—» ‡°º 4)

4.5 ‹µ„˜´ª°¥nµŠÄœ…o° 4 ‡¦¼Â¨³œ´„Á¦¸¥œnª¥„´œ­¦»ž Á¡Éº°Áž}œ„µ¦¥Êεªnµ ­´ÁŽ˜Ä—Ç šÉ¸Å¤nčnÁŽ˜
ªnµŠ…°Š R ™oµ¤¸…°Á…˜œÂ¨ªo ­´ÁŽ˜œ´Êœ¤…¸ °Á…˜œ‡nµœ°o ¥­»—Äœ R

4.6 ‡¦¼„¨µn ª™¹Š šœ¥· µ¤ 3-4 ¡¦°o ¤š´ÊŠ¥„˜´ª°¥µn Š°›· µ¥Ã—¥Äo„µ¦™µ¤˜° —Š´ ˜n°ÅžœÊ¸

˜´ª°¥nµŠ ‹Š®µ…°Á…˜¨nµŠ ¨³…°Á…˜¨nµŠ‡nµ¤µ„­—» …°Š S (™oµ¤)¸
1. S = (2,7) …………..(2 ¨³‹Îµœªœ‹¦·Šš»„‹ÎµœªœšÉ¸œo°¥„ªnµ 2 Áž}œ…°Á…˜

¨nµŠ…°Š S ¨³…°Á…˜¨nµŠ‡µn ¤µ„­—» ‡°º 2 )
2. S = {-2, 7, -5, 8, 9} …………..(-5 ¨³‹Îµœªœ‹¦·Šš»„‹Îµœªœš¸Éœo°¥„ªnµ -5 Áž}œ

…°Á…˜¨nµŠ…°Š S ¨³…°Á…˜¨µn Š‡nµ¤µ„­—» ‡°º -5 )
3. S = ( f,8 ) …………..(S Ť¤n …¸ °Á…˜¨nµŠ )

4.7 ‡¦¼°›· µ¥Ä®oœ´„Á¦¸¥œš¦µªnµ ­´‹¡‹œr„µ¦¤¸‡nµ…°Á…˜œœo°¥­»— Áž}œ­¤´˜·ž¦³„µ¦šÉ¸ 15
…°Š¦³‹Îµœªœ‹¦·Š ¨oª™µ¤¥Êεœ´„Á¦¸¥œÁ„ɸ¥ª„´­¤´˜·…°Š‹Îµœªœ‹¦·Š 14 …o°„n°œ®œoµœ¸Ê¤¸°³Å¦oµŠ Ä®o
œ´„Á¦¸¥œnª¥„´œ˜° ¨³°„œ´„Á¦¥¸ œªnµ ­¤˜´ …· °Š¦³‹µÎ œªœ‹¦·ŠšÊ´Š 15 …o°œÊ¸ Á¦¸¥„ªnµ ­´‹¡‹œr…°Š
¦³‹Îµœªœ‹¦·Š ŽÉ¹Š‹³…µ—…o°Ä—…o°®œ¹ÉŠÅ¤nŗo Á¡¦µ³™º°Áž}œÃ‡¦Š­¦oµŠ…°Š¦³‹Îµœªœ‹¦·Š ¨³‹³
œµÎ Şč¤o µ„Äœ‡–˜· «µ­˜¦r œÊ´ ­Š¼

4.8 ‡¦¼°„Ä®oœ´„Á¦¸¥œš¦µªnµšÉ¸Á¦¸¥œ¤µšÊ´Š®¤—˜´ÊŠÂ˜n‡µÂ¦„‹œ„¦³šÉ´Š¤µ™¹Š‡µœÊ¸ Áž}œ
„µ¦«¹„¬µÃ‡¦Š­¦oµŠ…°Š¦³‹Îµœªœ‹¦·Š „¨nµªÃ—¥­¦»ž ¦³‹Îµœªœ‹¦·ŠÁž}œ¦³ž¦³„°—oª¥ÁŽ˜
R ¨³ + , . šœ—ªo ¥ (R,+, . ) ¨³¤Ã¸ ‡¦Š­¦µo Š ‡º°­¤´˜·šÊŠ´ 15 …°o šœÉ¸ „´ Á¦¸¥œÅ—Áo ¦¥¸ œ¤µÂ¨oªœÉœ´ Á°ŠÂ¨oª
Ä®oœ´„Á¦¸¥œšµÎ Á°„­µ¦ f„®—´ š¸É 13 Ážœ} „µ¦µo œ

4.9 ‡¦Á¼ Œ¨¥Á°„­µ¦ f„®´—ÁŒ¡µ³…°o šÉ¸œ´„Á¦¸¥œ­Š­´¥ ššªœ ®¤—…o°­Š­´¥Â¨oª š—­°Á¦Éº°Š
nªŠÂ¨³„µ¦Â„o°­¤„µ¦ÄoÁª¨µ 30 œµš¸

5. ®¨nŠ„µ¦Á¦¸¥œ¦o¼
5.1 ®œŠ´ ­º°Á¦¸¥œ­µ¦³„µ¦Á¦¥¸ œ¦¼oÁ¡É¤· Á˜¤· ‡–˜· «µ­˜¦r Á¨n¤ 1 œÊ´ ¤´›¥¤«¹„¬µžše ¸É 4
5.2 Á°„­µ¦Âœ³ÂœªšµŠš¸É 13
5.3 Á°„­µ¦ f„®—´ š¸É 13
5.4 š—­° Á¦ºÉ°ŠnªŠÂ¨³„µ¦Â„o°­¤„µ¦

⌦ 101
⌦

6. „¦³ªœ„µ¦ª´—¨³ž¦³Á¤·œŸ¨ „µ¦ž¦³Á¤œ· Ÿ¨
1. œ„´ Á¦¸¥œ˜°‡µÎ ™µ¤Å—o™¼„˜o°ŠÁžœ} ­nªœ¤µ„
„µ¦ª—´ Ÿ¨ 2. œ´„Á¦¸¥œšµÎ ŗ™o ¼„˜o°Šž¦³¤µ– 80 %
1. ­´ŠÁ„˜‹µ„„µ¦˜°‡µÎ ™µ¤ 3. œ´„Á¦¥¸ œšµÎ ŗ™o „¼ ˜°o Šž¦³¤µ– 85 %
2. šÎµÁ°„­µ¦Âœ³ÂœªšµŠš¸É 13 4. œ´„Á¦¥¸ œšµÎ ŗo™„¼ ˜o°Šž¦³¤µ– 80 %
3. šµÎ Á°„­µ¦ f„®—´ š¸É 13
4. šÎµÂš—­°

7. ´œš¹„®¨´Š­°œ
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8. „‹· „¦¦¤Á­œ°Âœ³
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102 ⌫ ⌫  ⌦
 ⌫     ⌫  

⌦ 103
⌦

104 ⌫ ⌫  ⌦
 ⌫     ⌫  

Ÿœ„µ¦‹´—„µ¦Á¦¥¸ œ¦¼šo ¸É 14

Á¦°ºÉ Š „µ¦®µ¦¨Š˜ª´ Êœ´ ¤´›¥¤«„¹ ¬µžše ɸ 4
ª·µ ‡–·˜«µ­˜¦r Áª¨µ 4 ´ÉªÃ¤Š

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Ÿ¨„µ¦Á¦¸¥œ¦š¼o ‡É¸ µ—®ª´Š
°„‡ªµ¤®¤µ¥Â¨³Ä­o ´¨„´ ¬–r„µ¦®µ¦¨Š˜ª´ ŗo

1. ‹—» ž¦³­Š‡r„µ¦Á¦¸¥œ¦o¼ œ„´ Á¦¸¥œ­µ¤µ¦™
1.1 °„‡ªµ¤®¤µ¥Â¨³Äo­´¨„´ ¬–r„µ¦®µ¦¨Š˜´ªÅ—o
1.2 °„­¤´˜·…°Š„µ¦®µ¦¨Š˜ª´ ŗo
1.3 °„‡ªµ¤®¤µ¥…°Š‹ÎµœªœÁŒ¡µ³Å—o

2. œª‡ªµ¤‡—· ®¨´„
­¤´˜·šÉ¸­Îµ‡´…°Š¦³‹ÎµœªœÁ˜È¤ž¦³„µ¦®œ¹ÉŠ ŗo„n ­¤´˜·„µ¦®µ¦¨Š˜´ª „µ¦«¹„¬µ­¤´˜·

„µ¦®µ¦¨Š˜´ª‹³šÎµÄ®oÁ…oµÄ‹¦³‹ÎµœªœÁ˜È¤Å—o°¥nµŠ¨¹„ŽÊ¹Š ¨³Áž}œ¡Êºœ“µœÄœ„µ¦«¹„¬µ¦³‹Îµœªœ
Á˜È¤Äœ´œÊ ­¼Š˜°n Ş

3. Áœ°Êº ®µ­µ¦³
3.1 ­¤´˜„· µ¦®µ¦¨Š˜´ª
3.2 „µ¦‹µÎ œ„‹ÎµœªœÁ˜È¤Ã—¥­¤˜´ ·„µ¦®µ¦¨Š˜´ª

šœ·¥µ¤ Ä®o m ¨³ n Áž}œ‹µÎ œªœÁ˜¤È ¨³ n z 0 n ®µ¦ m ¨Š˜´ª„Șn°Á¤ºÉ° ¤¸‹ÎµœªœÁ˜È¤
c ššÉ¸ εĮo m nc

Á¦¥¸ „ n ªµn ˜´ª®µ¦ (Divisor) ®¦º°˜ª´ ž¦³„° (factor) ˜´ª®œÉŠ¹ …°Š m
Á¦¥¸ „ m ªnµÁžœ} ¡®‡» ¼– (Multiple) …°Š n

č­o ´¨„´ ¬–r n | m šœ “ n ®µ¦ m ¨Š˜´ª ”
n | m šœ “ n ®µ¦ m Ťn¨Š˜ª´ ”

⌦ 105
⌦

106 ⌫ ⌫  ⌦
 ⌫     ⌫  

‡¦¼°„œ„´ Á¦¸¥œªnµ Ĝ„¦–š¸ ¸É m,n ¨³ c Áž}œ‹µÎ œªœÁ˜È¤µŠ˜ª´ ¨³ n z 0 ™µo m
®µ¦—oª¥ n ŗoŸ¨®µ¦Áž}œ c ®¦º° m c ‹³„¨nµªªnµ n ®µ¦ m ¨Š˜´ª ˜nÁœÉº°Š‹µ„ÁŽ˜…°Š‹ÎµœªœÁ˜È¤

n

Ťn¤¸­¤´˜ž· —d …°Š„µ¦®µ¦ ‹¹Šœ·¥µ¤„µ¦®µ¦¨Š˜´ªÃ—¥Äo m nc
4.2 ‡¦Â¼ ‹„Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 14 Ä®œo „´ Á¦¥¸ œ‹´ ‡„n¼ œ´ «„¹ ¬µ ¨³˜°‡µÎ ™µ¤Ã—¥„µ¦Á˜¤·

‡Îµ˜°ÄœµŠ­ªn œšÁɸ ªœo Ūo ¨³™µ¤œ„´ Á¦¥¸ œªµn ™oµ n ®µ¦ m Ťn¨Š˜ª´ ‡ª¦‹³­¦»ž°¥µn ŠÅ¦ ŽÉŠ¹ œ´„Á¦¸¥œ
‡ª¦­¦»žÅ—ªo nµ

“ n ®µ¦ m Ť¨n Š˜ª´ „Ș°n Á¤É°º Ť¤n ‹¸ µÎ œªœÁ˜¤È c Ä—Ç š¸ÉšµÎ Ä®o m nc ”
4.3 Ä®oœ´„Á¦¸¥œ«¹„¬µ­¤´˜·„µ¦®µ¦¨Š˜´ª‹µ„Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 14 ¨oª‡¦¼Â¨³œ´„Á¦¸¥œ
ªn ¥„œ´ ¡­· ‹¼ œr ×¥„µ¦™µ¤Ä®oœ„´ Á¦¥¸ œnª¥„œ´ ˜°—´Š˜°n ޜʸ

š§¬‘¸ š 1 Á¤É°º a,b  , ×¥šÉ¸ a,b z 0
™µo a | b ¨³ b | c ¨oª a | c

¡­· ‹¼ œr ­¤¤»˜· a | b ¨³ b | c ‹³¤‹¸ 圪œÁ˜È¤ x š¸šÉ εĮo b = ax
¨³ ¤‹¸ µÎ œªœÁ˜È¤ y š¸É c = by
—´Šœ´œÊ = (ax)y

š§¬‘¸ š 2 = a(xy)
¡­· ‹¼ œr
Áœ°Éº Š‹µ„ xy Áž}œ‹µÎ œªœÁ˜¤È
—Š´ œ´Êœ a | c
™oµ a ¨³ b Áž}œ‹ÎµœªœÁ˜È¤ª„ ŽŠ¹É a | b ‹³Å—o a d b
­¤¤»˜· a | b
‹³¤¸‹ÎµœªœÁ˜¤È x ššÉ¸ µÎ Ä®o b = ax
ÁœÉº°Š‹µ„ a ¨³ b Áž}œ‹ÎµœªœÁ˜¤È ª„ ‹³Å—o x Áž}œ‹µÎ œªœÁ˜È¤ª„
Œ³œÊœ´ x t 1

ax t a (a  I )

—´Šœ´œÊ b t a

š§¬‘¸š 3 ™oµ a,b,c  I ¨³ a z 0 ×¥š¸É a | b ¨³ a | c
¡·­¼‹œr
¨oª a | bx cy Á¤°ºÉ x ¨³ y Ážœ} ‹ÎµœªœÁ˜¤È ė Ç

­¤¤˜» · a | b ¨³ a | c

‹³¤¸ m  I ššÉ¸ εĮo b = am …….( 1 )

¨³¤¸ n  I š¸šÉ εĮo c = an ……( 2 )

—Š´ œÊ´œ ( 1 )u x ‹³Å—o bx amx ..….( 3 )

⌦ 107
⌦

108 ⌫ ⌫  ⌦
 ⌫     ⌫  

‹µ„…µo Š˜oœ ­¦ž» Áž}œš§¬‘¸ šÅ——o Š´ œ¸Ê

š§¬‘¸š 4 š§¬‘¸š®¨´„¤¼¨šµŠÁ¨…‡–·˜ (The Fundamental Theorem of Arithmetic)

š»„‹ÎµœªœÁ˜È¤ª„ n šÉ¸¤µ„„ªnµ 1 ‹³­µ¤µ¦™Â¥„˜´ªž¦³„°ÁŒ¡µ³—´Š˜n°ÅžœÊ¸ ŗo¦¼žÁ—¸¥ª‡º°

n P C1 ˜ P C2 ˜ P C3 ... ˜ P Ck
1 2 3 k

Ž¹ŠÉ P1 P2 P3 ... Pk ¨³ p š„» ˜´ªÁžœ} ‹ÎµœªœÁŒ¡µ³ª„ ¨³ c š»„˜ª´ Áž}œ‹ÎµœªœÁ˜È¤ª„

4.9 ­n»¤Ä®oœ´„Á¦¸¥œ 2 ‡œ °°„¤µÁ…¸¥œ‹Îµœªœš¸É‡¦¼„ε®œ—Ä®oĜ¦¼ž˜´ªž¦³„°ÁŒ¡µ³˜µ¤š§¬‘¸

š 4 —Š´ œÊ¸

1. 720 œ´„Á¦¸¥œ‡ª¦Á…¥¸ œÅ—Áo žœ} 720 = 24 u 32 u 5

2. 4725 œ„´ Á¦¸¥œ‡ª¦Á…¥¸ œÅ—Áo ž}œ 4725 = 33 u 52 u 7

4.10 Áž—d ð„µ­Ä®œo ´„Á¦¥¸ œšÉ­¸ Š­´¥Ž´„™µ¤ ¨ªo Ä®ošµÎ  „f ®´— 3.1 … Ážœ} „µ¦oµœ

5. ®¨nŠ„µ¦Á¦¥¸ œ¦¼o
5.1 Á°„­µ¦Âœ³ÂœªšµŠš¸É 14
5.2 Á°„­µ¦ f„®´—š¸É 14
5.3 ®œŠ´ ­°º Á¦¥¸ œ­µ¦³„µ¦Á¦¥¸ œ¦¼Áo ¡·¤É Á˜·¤ ‡–·˜«µ­˜¦Ár ¨¤n 1 ´Êœ ¤.4

6. „µ¦ª—´ ¨³„µ¦ž¦³Á¤œ· Ÿ¨

„µ¦ª—´ Ÿ¨ „µ¦ž¦³Á¤·œŸ¨

1. ­´ŠÁ„˜‹µ„„µ¦˜°‡µÎ ™µ¤ 1. œ´„Á¦¥¸ œ˜°‡µÎ ™µ¤Å—o™„¼ ˜°o ŠÁž}œ­ªn œ¤µ„

2. ­´ŠÁ„˜‹µ„‡ªµ¤­œÄ‹ 2. œ´„Á¦¥¸ œ­œÄ‹Â¨³˜Š´Ê ċÁ¦¸¥œ

3. šµÎ ˚¥rÁ°„­µ¦ f„®—´ š¸É 14 3. œ´„Á¦¸¥œšµÎ ŗ™o „¼ ˜o°Šž¦³¤µ– 80 %

4. šÎµÃ‹š¥Âr  f„®´— 3.1 „ Ĝ®œ´Š­º° 4. œ´„Á¦¸¥œšÎµÅ—™o ¼„˜°o Šž¦³¤µ– 80 %

Á¦¥¸ œ­µ¦³„µ¦Á¦¸¥œ¦¼Áo ¡·¤É Á˜¤·

7. ´œš¹„®¨Š´ „µ¦­°œ
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8. „·‹„¦¦¤Á­œ°Âœ³
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⌦ 109
⌦

Á°„­µ¦Âœ³ÂœªšµŠš¸É 14

‡µÎ Âʸ ‹Š Ä®œo „´ Á¦¸¥œ«„¹ ¬µœ¥· µ¤Â¨³š§¬‘¸˜µn ŠÇ …µo Š¨nµŠœ¸Ê ¨oªÁ˜·¤‡µÎ ˜°šÉÁ¸ ªoœÅªo

œ·¥µ¤ Ä®o m ¨³ n Áž}œ‹ÎµœªœÁ˜È¤ ¨³ n z 0 n ®µ¦ m ¨Š˜´ª„Șn°Á¤ºÉ° ¤‹¸ 圪œÁ˜È¤ c

ŽÉ¹Š m nc

Á¦¸¥„ n ªµn ˜ª´ ®µ¦ (Divisor) ®¦°º ˜´ªž¦³„° (factor) ˜´ª®œŠÉ¹ …°Š m
Á¦¸¥„ m ªµn ¡®‡» –¼ (Multiple) …°Š n
č­o ´¨„´ ¬–r n | m šœ n ®µ¦ m ¨Š˜ª´

n | m šœ n ®µ¦ m Ťn¨Š˜ª´

3 | 12 (°nµœªµn 3 ®µ¦ 12 ¨Š˜ª´ ) Á¡¦µ³ªµn ¤¸‹µÎ œªœÁ˜È¤ 4 ŽÉŠ¹ šÎµÄ®o 12 3 4
3 | 12 Á¡¦µ³ªµn ¤¸‹ÎµœªœÁ˜È¤ -4 ŽŠ¹É šµÎ Ä®o 12 3 4
5 | 15 Á¡¦µ³ªµn _____________________________________________________
4 | 16 Á¡¦µ³ªnµ_____________________________________________________

6 | 0 Á¡¦µ³ªnµ_____________________________________________________
5 | 3 Á¡¦µ³ªµn _____________________________________________________

2 | 5 Á¡¦µ³ªµn _____________________________________________________

­¤˜´ „· µ¦®µ¦¨Š˜´ª

š§¬‘¸ š 1 Á¤É°º a, b  I ¨³ a, b z 0 ™µo a | b ¨³ b | c ‹³Å—o a | c

š§¬‘¸š 2 ™oµ a ¨³ b Áž}œ‹µÎ œªœÁ˜È¤ª„ ŽŠÉ¹ a | b ‹³Å—o a d b

š§¬‘¸ š 3 ™µo a, b, c  I ¨³ a z 0 ×¥š¸É a | b ¨³ a | c ‹³Å—o a | bx cy Á¤Éº°

x, y I

Ÿ¨‹µ„š§¬‘¸ š 3

™oµ a | b ¨³ a | c ‹³Å—o

1. a | b c
2. a | bx c , x  I
3. a | bx c , x  I

‹³Á¦¥¸ „œ·¡‹œrĜ¦¼ž bx cy ªµn Ÿ¨¦ª¤ÁŠ· Á­oœ (linear combination)…°Š b ¨³ c

110 ⌫ ⌫  ⌦
 ⌫     ⌫  

Á°„­µ¦ „f ®—´ šÉ¸ 14

°ºÉ ……………………………………………………œ´Ê ………………………..Á¨…š…ɸ ………………

‡Îµ­ÉŠ´ …°o ‡ªµ¤ÄœÂ˜¨n ³…°o ˜n°Åžœ¸ÁÊ žœ} ‹¦·Š®¦°º ÁšÈ‹
……..1. ™µo a|1 ¨oª a = 1
……..2. ™µo a| (b+c) ¨oª a|b ®¦º° a|c
……..3. ™oµ a|b ¨ªo a|b2
……..4. ™µo a|b ¨³ a|c ¨ªo a|(b+c)
……..5. ‹µÎ œªœ˜°n Şœ¸Ê 179, 209, 327 Áž}œ‹µÎ œªœÁŒ¡µ³š»„‹Îµœªœ
……..6. ™oµ a ®µ¦ b ¨Š˜ª´ ¨³ b ®µ¦ c Ť¨n Š˜ª´ ¨ªo a ®µ¦ c Ť¨n Š˜ª´
……..7. ™oµ a|b ¨³ a|c ¨ªo a|(bx-cy) Á¤º°É x, y Áž}œ‹ÎµœªœÄ—Ç
……..8. ™oµ a|b ¨³ a|c ¨oª a|(bx+cy) Á¤º°É x, y Ážœ} ‹µÎ œªœÁ˜¤È ėÇ
……..9. ™µo a| c ¨³ b| c ¨ªo ab| c
…….10. ™oµ ab| c ¨oª a| c ¨³ b| c

⌦ 111
⌦

Ÿœ„µ¦‹´—„µ¦Á¦¸¥œ¦¼šo ɸ 15

Á¦°ºÉ Š ­¤´˜…· °Š‹µÎ œªœÁ˜¤È (˜°n ) ´Êœ¤´›¥¤«¹„¬µžše ɸ 4
ª·µ ‡–˜· «µ­˜¦r Áª¨µ 4 ªÉ´ äŠ

***********************************************************************************

Ÿ¨„µ¦Á¦¸¥œ¦šo¼ ¸É‡µ—®ª´Š
œµÎ š§¬’¸š…Ê´œ˜°œª›· „¸ µ¦®µ¦ÅžÄÅo —o

1. ‹»—ž¦³­Š‡„r µ¦Á¦¸¥œ¦¼o œ´„Á¦¥¸ œ­µ¤µ¦™
1.1 °„Ä‹‡ªµ¤Â¨³‡ªµ¤®¤µ¥…°Šš§¬‘¸š…Ê´œ˜°œª›· „¸ µ¦®µ¦Å—o
1.2 °„œ¥· µ¤…°Š‹µÎ œªœ‡¼n¨³‹Îµœªœ‡É¸ ¨³œµÎ ŞčÅo —o

2. œª‡ªµ¤‡·—®¨´„
­¤´˜·šÉ¸­Îµ‡´…°Š¦³‹ÎµœªœÁ˜È¤°¸„ž¦³„µ¦®œÉ¹Š ŗo„n …Ê´œ˜°œª·›¸„µ¦®µ¦ „µ¦«¹„¬µš§¬‘¸

š…´Êœ˜°œª·›¸„µ¦®µ¦ ‹³œÎµÅž­¼n„µ¦œ·¥µ¤ ‹Îµœªœ‡¼n ‹Îµœªœ‡É¸ ˜´ª®µ¦¦nª¤¤µ„¨³˜´ª‡¼–¦nª¤œo°¥
¨³‹³Áž}œ¡Êºœ“µœÄœ„µ¦«¹„¬µª·µ¡¸‡–·˜œµ¤›¦¦¤˜n°Åž

3. ÁœºÊ°®µ­µ¦³
…œ´Ê ˜°œª›· ¸„µ¦®µ¦ (Division Algorithm)

š§¬‘¸š 5 …Êœ´ ˜°œª›· „¸ µ¦®µ¦
™µo m ¨³ n Áž}œ‹ÎµœªœÁ˜¤È ×¥š¸É n z 0 ‹³¤‹¸ µÎ œªœÁ˜¤È q ¨³ r »—Á—¥¸ ª ŽÉŠ¹
m nq r ×¥š¸É 0 d r | n |
Á¦¥¸ „ q ªµn Ÿ¨®µ¦(quotient) Ánœ 24 = 4(6) + 0 , -20 = 6(-4) + 4
¨³ r ªµn Á«¬Á®¨º° (remainder)

šœ¥· µ¤
‹ÎµœªœÁ˜È¤ a Áž}œ‹µÎ œªœ‡¼n „˜È n°Á¤°Éº ­µ¤µ¦™Á…¸¥œ a = 2n Á¤°ºÉ n Ážœ} ‹ÎµœªœÁ˜¤È
‹ÎµœªœÁ˜È¤ a Áž}œ‹Îµœªœ‡¸É „˜È n°Á¤ºÉ°­µ¤µ¦™Á…¸¥œ a = 2n+1 Á¤°Éº n Ážœ} ‹ÎµœªœÁ˜¤È

112 ⌫ ⌫  ⌦
 ⌫     ⌫  

‹µ„œ¥· µ¤ 0 Ážœ} ‹Îµœªœ‡nÁ¼ ¡¦µ³ 0 = 2(0)

š§¬‘¸ š 6 Ä®o b Áž}œ‹µÎ œªœÁ˜È¤š¤É¸ µ„„ªµn 1 ‹µÎ œªœÁ˜È¤ª„ n Ä—Ç ­µ¤µ¦™Á…¸¥œ
Ĝ¦ž¼ „µ¦„¦³‹µ¥“µœ b ŗÁo žœ}

n = akbk + ak-1bk-1 + ak-2bk-2 +…+ a1b + a0
Á¤°Éº k Ážœ} ‹µÎ œªœÁ˜¤È ¨³ a0 , a1 , a2 ,…, ak Ážœ} ‹µÎ œªœÁ˜¤È šÉŸ ¤nÁžœ}
¨Â¨³Å¤nœ°o ¥„ªnµ b ¨³ ak z 0

4. „¦³ªœ„µ¦‹—´ „µ¦Á¦¥¸ œ¦o¼

4.1 ‡¦¼„ε®œ—‹Îµœªœ ¨oªÄ®oœ´„Á¦¸¥œ®µŸ¨®µ¦ ¨³Á«¬‹µ„„µ¦®µ¦ —´Šœ¸Ê (ŽÉ¹Šœ´„Á¦¸¥œ‡ª¦˜°

ŗ—o ´Šš¸É­—ŠÅªÄo œªŠÁ¨È )

1. 20 ®µ¦—ªo ¥ 5 (Ÿ¨®µ¦ 4 ,Á«¬ 0 )

2. 27 ®µ¦—oª¥ 4 (Ÿ¨®µ¦ 6 ,Á«¬ 3 )

3. -24 ®µ¦—ªo ¥ 5 (Ÿ¨®µ¦ 4 ,Á«¬ 4 )

4. 74 ®µ¦—oª¥ –3 (Ÿ¨®µ¦ 24 ,Á«¬ 2 )

‹µ„Ÿ¨®µ¦š¸ÅÉ —oĜ…´ÊœœÎµ ‡¦¼Êĸ ®oœ„´ Á¦¸¥œÁ®Èœªnµ

˜´ª˜´ŠÊ ˜ª´ ®µ¦ u Ÿ¨®µ¦ Á«¬
‹µ„…o° 1 Á…¥¸ œÅ—oÁžœ} 20 5 4

‹µ„…°o 2 Á…¥¸ œÅ—oÁžœ} 27 4 6 3

‹µ„…°o 3 Á…¥¸ œÅ—Áo ž}œ 24 5 4 4

‹µ„…o° 4 Á…¸¥œÅ—Áo žœ} 74 3 24 2

‡¦¼™µ¤œ´„Á¦¸¥œªnµ ‹µ„…o° 3 ™oµ˜o°Š„µ¦Ä®oÁ«¬Áž}œ‹Îµœªœª„ ‹³Á…¸¥œÅ—o°¥nµŠÅ¦

œ´„Á¦¸¥œ‡ª¦˜°Å—oªnµ 24 5 5 1

4.2 Ÿ¨‹µ„…°o š¸É 1 ‡¦Â¼ ¨³œ„´ Á¦¥¸ œªn ¥„œ´ ­¦»žÁž}œš§¬‘¸ š Á„¸¥É ª„´ …œ´Ê ˜°œª·›„¸ µ¦®µ¦Å——o ´ŠœÊ¸

š§¬‘¸š 5 …œÊ´ ˜°œª·›„¸ µ¦®µ¦
Ä®o m ¨³ n Ážœ} ‹µÎ œªœÁ˜¤È n z 0 ‹³¤‹¸ µÎ œªœÁ˜¤È q ¨³ r »—Á—¸¥ª ŽÉŠ¹

m nq r ×¥š¸É 0 d r | n |
Á¦¥¸ „ q ªnµŸ¨®µ¦ ¨³ r ªnµÁ«¬

4.3 Ä®oœ„´ Á¦¥¸ œšµÎ  f„®—´ 3.2 …o° 1 , 2 ‹µ„®œ´Š­°º Á¦¸¥œ­µ¦³„µ¦Á¦¥¸ œ¦Áo¼ ¡¤É· Á˜¤·
4.4 ‡¦¼„¨nµª™¹Š‹Îµœªœ‡n¼Â¨³‹Îµœªœ‡¸É×¥„µ¦™µ¤‡ªµ¤®¤µ¥ Ä®oœ´„Á¦¸¥œnª¥„´œ˜° Ž¹ÉŠ
œ´„Á¦¸¥œ°µ‹‹³˜°ªµn ‹Îµœªœ‡¼®n ¤µ¥™Š¹ ‹ÎµœªœšÉ¸ 2 ®µ¦¨Š˜´ª ¨³‹Îµœªœ‡¸É®¤µ¥™¹Š‹ÎµœªœšÉ¸ 2 ®µ¦Å¤n¨Š

⌦ 113
⌦

˜´ª ‡¦¼Ê¸Âœ³Ä®oœ´„Á¦¸¥œÁ®Èœªnµ ‹µ„š§¬’¸š 5 Ĝ…o°š¸É 5.2 ­µ¤µ¦™œÎµÅžÄoĜ„µ¦œ·¥µ¤‹Îµœªœ‡n¼
‹µÎ œªœ‡¸É Á¤°ºÉ 2 Ážœ} ˜´ª®µ¦Å—o —´Š˜n°ÅžœÊ¸

šœ¥· µ¤
‹ÎµœªœÁ˜È¤ a ‹³Ážœ} ‹Îµœªœ‡¼n „˜È °n Á¤Éº°­µ¤µ¦™Á…¥¸ œ a = 2n Á¤°ºÉ n Ážœ} ‹ÎµœªœÁ˜È¤
‹ÎµœªœÁ˜È¤ a ‹³Ážœ} ‹µÎ œªœ‡¸É „˜È °n Á¤º°É ­µ¤µ¦™Á…¥¸ œ a = 2n+1 Á¤°Éº n Áž}œ‹ÎµœªœÁ˜¤È

4.5 ‡¦¼¥„˜´ª°¥nµŠÃ‹š¥r„µ¦¡·­¼‹œr 1 …o° ×¥„µ¦™µ¤Ä®oœ´„Á¦¸¥œnª¥„´œ˜° ‡¦¼Á…¸¥œœ
„¦³—µœ—ε —Š´ ˜°n Şœ¸Ê

˜´ª°¥nµŠ ‹ŠÂ­—Šªµn ™µo x Áž}œ‹µÎ œªœ‡Âɸ ¨oª 4|(x2-1)
ª·›¸šÎµ Ä®o x = 2n + 1 Á¤º°É n Áž}œ‹ÎµœªœÁ˜È¤
‹³Å—o x2 = (2n + 1 )2
= 4n2+4n+1

x2-1 = 4n2+4n (ª„—ªo ¥ –1 š´ÊŠ­°Š…oµŠ)
= 4 (n2+n)

Á¡¦µ³ªµn n Ážœ} ‹ÎµœªœÁ˜¤È —´ŠœÊœ´ (n2+n) Áž}œ‹µÎ œªœÁ˜È¤
‹µ„­¤´˜·…°Š‹ÎµœªœÁ˜¤È ‹³Å—o 4|(x2-1)

4.6 ‡¦¼¥„˜´ª°¥nµŠ„µ¦œÎµ…´Êœ˜°œª·›¸„µ¦ÅžÄož¦³Ã¥œr ×¥„µ¦Á…¸¥œÃ‹š¥rœ„¦³—µœ—娳

čªo ›· ¸„µ¦°›·µ¥ ™µ¤˜°

˜´ª°¥µn Š ‹Š®µ‹µÎ œªœÁ˜È¤ª„š´ŠÊ ®¤—š®É¸ µ¦ 417 ¨³ 390 ¨oª¤Á¸ «¬Á®¨°º Ášnµ„œ´

ª›· ¸šµÎ Ä®o x  I x ®µ¦ 417 ¨³ 390 ¤Á¸ «¬Á®¨°º Ášnµ„´ r

—Š´ œÊ´œ 417 kx r ; k I...............(1)

390 mx r ; m I...............(2)

(1) – (2) 27 = (k-m)x

Áœ°ºÉ Š‹µ„ (k m)  I —Š´ œÊ´œ x | 27
Á¡¦µ³Œ³œœ´Ê ‡µn x šÁɸ žœ} Şŗ‡o °º 1,3,9,27
‹µ„„µ¦˜¦ª‹­° 1 ¨³ 3 ®µ¦ 417 ¨³ 390 ¨Š˜ª´ (Á«¬Ášµn „´ 0)

9 ®µ¦ 417 ¨³ 390 ˜nµŠ¤Á¸ «¬Á®¨º°Ášnµ„´ 3
27 ®µ¦ 417 ¨³ 390 ˜µn Š¤¸Á«¬Á®¨º°Ášnµ„´ 12
—´ŠœœÊ´ ‹ÎµœªœÁ˜¤È ª„šŠ´Ê ®¤—šÉ¸®µ¦ 471 ¨³ 390 ¤¸Á«¬Á®¨º°Ášµn „´œ‡º° 1,3,9 ¨³ 27
®¨´Š‹µ„š»„‡œ®¤—…o°­Š­¥´ ¨ªo Ä®šo µÎ  „f ®´— 3.2 …°o 2 Ážœ} „µ¦µo œ
4.7 ‡¦¼¥„˜´ª°¥nµŠ‹Îµœªœ 368 ,45802 ¨³Ä®oœ´„Á¦¸¥œÁ…¸¥œÄœ¦¼ž„µ¦„¦³‹µ¥“µœ­·œ´„Á¦¸¥œ
‡ª¦Á…¸¥œÅ—oªµn

368 (3x102 ) (6x10) 8
45802 (4x104 ) (5x103 ) (8x102 ) (0x10) 2

114 ⌫ ⌫  ⌦
 ⌫     ⌫  

4.8 ‡¦°¼ ›·µ¥ªnµÃ—¥šÉ´ªÅž Á¦µ­µ¤µ¦™Á…¥¸ œ‹µÎ œªœÁ˜¤È ª„Äœ¦ž¼ „µ¦„¦³‹µ¥…°Š“µœ˜nµŠÇ ŗo
—Š´ š§¬’˜¸ n°Åžœ¸Ê

š§¬‘¸ š 6 Ä®o b Áž}œ‹ÎµœªœÁ˜È¤š¤¸É µ„„ªnµ 1 ‹ÎµœªœÁ˜¤È ª„ n Ä—Ç ­µ¤µ¦™Á…¸¥œÄœ¦ž¼ „µ¦

„¦³‹µ¥“µœ b ŗÁo žœ}
b = akbk + ak-1bk-1 + ak-2bk-2 +…+ a1b + a0

Á¤É°º k Áž}œ‹ÎµœªœÁ˜È¤Â¨³ a0 , a1 , a2 ,…, ak Ážœ} ‹µÎ œªœÁ˜¤È šÅɸ ¤nÁž}œ¨Â¨³Å¤nœ°o ¥„ªnµ b ¨³

ak z 0

4.9 ‡¦¼Â¨³œ´„Á¦¸¥œ¦nª¤„´œ¡·­¼‹œrץčo…Ê´œ˜°œª·›¸„µ¦®µ¦ Á¤ºÉ° ®µ¦ n —oª¥ b ŗoŸ¨®µ¦ q0
¨³Á®¨°º Á«¬ a0 Á…¥¸ œÅ——o ´Š¦ž¼

n bq0 a0 0 d a0 b (1)

®µ¦ q0 —ªo ¥ b ‹³Å—o

q0 bq1 a1 0 d a1 b (2)

šÎµÁnœœ¸ÅÊ žÁ¦°ºÉ ¥ Ç ‹œ„¦³šÉ´ŠŸ¨®µ¦Áž}œ 0

q1 bq2 a2 0 d a2 b
q2 bq3 a3 0 d a3 b

. .

..

..

qk 1 bqk ak 0 d ak b (3)

Á¡¦µ³ªnµ n > q0 > q1 > q2 > … t 0 Áž}œ¨Îµ—´…°Š‹ÎµœªœÁ˜È¤Å¤nÁž}œ¨šÉ¸¤¸‡nµ¨—¨Š —´Šœ´Êœ

…´œÊ ˜°œ„µ¦®µ¦…oµŠ˜œo ˜o°Š­·Êœ­»—×¥šŸÉ¸ ¨®µ¦˜´ª­—» šoµ¥Áž}œ«¼œ¥r —Š´ ­¤„µ¦šÉ¸ (1)

n bq0 a0

b(bq1 a1 ) a0
b 2q1 a1b a0

šœÅžÁ¦°Éº ¥ Ç

n b2 (bq2 a2 ) a1b a 0

b3q2 b 2a2 a1b a0

˜‹n µ„ (3) qk-1 = ak
—Š´ œ´œÊ n = akbk + ak-1bk-1 + ak-2bk-2 +…+ a1b + a0
˜´ª°¥nµŠ ‹ŠÁ…¥¸ œ 96 Ĝ¦ž¼ „¦³‹µ¥“µœ 5

ª·›¸šµÎ 96 = (5x19) + 1

= (5x(5x3) + 4) + 1
= (52x3) + 5x4 +1
96 = (3x52) + (4x5) + 1 = (341)5

⌦ 115
⌦

˜´ª°¥nµŠ ‹ŠÁ…¥¸ œ 25 Ĝ¦¼ž“µœ 2
ª·›š¸ µÎ 25 = 2(12) + 1
12 = 2(6) + 0

6 = 2(3) + 1

3 = 2(1) + 1

1 = 2(0) + 1
—´ŠœÊœ´ 25 = (11001)2
4.10 Ä®œo „´ Á¦¥¸ œšµÎ  f„®´— 3.2 …o° 3 ‹µ„®œ´Š­º°ÂÁ¦¸¥œ­µ¦³„µ¦Á¦¸¥œ¦¼oÁ¡·É¤Á˜·¤ ®œoµ 103
¨³Ä®œo ´„Á¦¥¸ œš»„‡œ­¤¤˜Á· ¨… 3 ®¨„´ ¨³Áž¨¸¥É œÁž}œ“µœ 4 ¨³ “µœ 6

5. ®¨Šn „µ¦Á¦¸¥œ¦o¼
®œŠ´ ­º°Á¦¥¸ œ­µ¦³„µ¦Á¦¸¥œ¦¼oÁ¡¤·É Á˜¤· ‡–˜· «µ­˜¦r Á¨n¤ 1 œÊ´ ¤›´ ¥¤«„¹ ¬µžše ɸ 4 …°Š ­­ªš.

6. „¦³ªœ„µ¦ª´—¨³ž¦³Á¤œ· Ÿ¨

„µ¦ª—´ Ÿ¨ „µ¦ž¦³Á¤œ· Ÿ¨
1. ­Š´ Á„˜‹µ„„µ¦˜°‡Îµ™µ¤ 1. œ„´ Á¦¥¸ œ˜°‡Îµ™µ¤Å—o™¼„˜°o ŠÁž}œ­ªn œ¤µ„
2. ­Š´ Á„˜‹µ„‡ªµ¤­œÄ‹ 2. œ„´ Á¦¸¥œ­œÄ‹Â¨³˜ÊŠ´ ċÁ¦¥¸ œ
3. šÎµÃ‹š¥rÁ°„­µ¦ „f ®—´ 23.1, 23.2 3. œ„´ Á¦¥¸ œšÎµÅ—™o „¼ ˜o°Šž¦³¤µ– 80 %
4. šÎµÃ‹š¥r „f ®—´ 2.11 Ĝ®œ´Š­°º 4. œ´„Á¦¸¥œšµÎ ŗo™„¼ ˜o°Šž¦³¤µ– 80 %

Á¦¥¸ œ² 5. œ„´ Á¦¸¥œšÎµÅ—™o ¼„˜°o Šž¦³¤µ– 80 %
5. šµÎ š—­°Á¦ºÉ°Š­¤´˜…· °Š‹ÎµœªœÁ˜È¤

7. ´œš„¹ ®¨Š´ ­°œ
.......................................................................................................................................................................
.......................................................................................................................................................................
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8. „·‹„¦¦¤Á­œ°Âœ³
.......................................................................................................................................................................
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116 ⌫ ⌫  ⌦
 ⌫     ⌫  

Ÿœ„µ¦‹´—„µ¦Á¦¥¸ œ¦¼ošÉ¸ 16

Á¦º°É Š ˜ª´ ®µ¦¦nª¤¤µ„ (®.¦.¤.) ´œÊ ¤›´ ¥¤«„¹ ¬µžeš¸É 4
ª· µ ‡–·˜«µ­˜¦r Áª¨µ 4 ªÉ´ äŠ

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Ÿ¨„µ¦Á¦¥¸ œ¦šo¼ ¸É‡µ—®ªŠ´
°„‡ªµ¤®¤µ¥ č­o ´¨´„¬–r …°Š®.¦.¤. ¨³œµÎ ‡ªµ¤¦Á¼o ¦°ºÉ Š ®.¦.¤. Şčoŗo

1. ‹»—ž¦³­Š‡r„µ¦Á¦¸¥œ¦o¼ œ´„Á¦¸¥œ­µ¤µ¦™
1.1 °„‡ªµ¤®¤µ¥Â¨³Äo­´¨´„¬–r…°Š ®.¦.¤. ŗo
1.2 ®µ ®.¦.¤. ץčo…Êœ´ ˜°œª›· ¸…°Š¥‡¼ ¨—· ŗo
1.3 œÎµ‡ªµ¤¦Á¼o ¦º°É Š ®.¦.¤. Şčo„o˚¥žr { ®µÅ—o

2. œª‡ªµ¤‡·—®¨„´
2.1 ‹ÎµœªœÁ˜È¤ª„ d ‹³Á¦¸¥„ªnµÁž}œ˜´ª®µ¦¦nª¤¤µ„(®.¦.¤.) …°Š‹ÎµœªœÁ˜È¤ a ¨³ b Ž¹ÉŠ °¥nµŠ

œo°¥ 1 ˜ª´ šÉŸ ¤nÁžœ} «œ¼ ¥r„˜È °n Á¤ºÉ°
1. d | a ¨³ d | b
2. ­µÎ ®¦´‹ÎµœªœÁ˜È¤ c š»„˜ª´ ™oµ c | a ¨³ c | b ¨ªo c | d
šœ ®.¦.¤. …°Š a ¨³ b —oª¥ (a, b)

2.2 …´Êœ˜°œª›· ®¸ µ ®.¦.¤. …°Š‹µÎ œªœÁ˜È¤ a ¨³ b …°Š¥‡» ¨·—
Á…¸¥œ a, b Ä®o°¥Ä¼n œ¦ž¼ a = q1b + r1 ; 0 d r1 d b
Á…¸¥œ b Ä®o°¥Än¼ œ¦¼ž b = q2 r1 + r2 ; 0 d r2 d r1
Á…¸¥œ r1 Ä®°o ¥Än¼ œ¦¼ž r1 = q3r2 + r3 ; 0 d r3 d r2
šÎµÁœn œÊ¸ÅžÁ¦°Éº ¥ Ç ‹œ„ªnµ‹³Å—o rn+1 = 0
‹³Å—o ®.¦.¤. …°Š a ¨³ b = rn

2.3 ™µo a , b  I Ž¹ÉŠ˜µn ŠÅ¤nÁšµn „´ «œ¼ ¥r ¨³ d = (a, b) ¨ªo ‹³Å—oªnµ¤‹¸ µÎ œªœÁ˜¤È x ¨³ y
ŽÉŠ¹ d = ax + by

2.4 ‹ÎµœªœÁ˜È¤ a ¨³ b ‹³Á¦¸¥„ªnµÁž}œ‹ÎµœªœÁŒ¡µ³­´¤¡´š›r (relatively prime) „Șn°Á¤Éº°

(a, b) = 1

⌦ 117
⌦

3. Áœ°Êº ®µ­µ¦³
3.1 ˜ª´ ®µ¦¦nª¤

šœ·¥µ¤ „µÎ ®œ— a ¨³ b Ážœ} ‹ÎµœªœÁ˜¤È Á¦¥¸ „‹ÎµœªœÁ˜È¤ c š¸­É µ¤µ¦™®µ¦š´ÊŠ a
¨³ b ¨Š˜´ª ªnµÁž}œ˜ª´ ®µ¦¦ªn ¤ …°Š a ¨³ b

3.2 ˜´ª®µ¦¦ªn ¤¤µ„

šœ¥· µ¤ Ä®o a ¨³ b Áž}œ‹ÎµœªœÁ˜È¤ ×¥šÉ¸ a ¨³ b ŤnÁž}œ«œ¼ ¥¡r ¦°o ¤„œ´
‹µÎ œªœÁ˜¤È ª„ d š¤É¸ ¸‡µn ¤µ„š¸É­»— ŽÉŠ¹ d | a ¨³ d | b Á¦¥¸ „ªµn Ážœ} ˜ª´ ®µ¦¦nª¤¤µ„
(®.¦.¤.) …°Š a ¨³ b č­o ´ ¨´„¬–r (a, b) šœ ®.¦.¤. …°Š a ¨³ b

3.3 „µ¦®µ˜ª´ ®µ¦¦nª¤¤µ„ץč…o ´Êœ˜°œª›· …¸ °Š¥‡» ¨—·

š§¬‘¸š 7 …´œÊ ˜°œª›· …¸ °Š¥‡» ¨·— (Euclidean Algorithm)
„µÎ ®œ—Ä®o a ¨³ b Áž}œ‹ÎµœªœÁ˜È¤ª„ ×¥šÉ¸ b < a
ץč…o ´Êœ˜°œª·›„¸ µ¦®µ¦ÅžÁ¦É°º ¥ Ç ‹³Å—ªo nµ

a = bq1 + r1 ; 0 < r1 < b
b = r1q2 + r2 ; 0 < r2 < r1
r1 = r2q3 + r3 ; 0 < r3 < r2

.

.

.

rk-2 = rk-1qk + rk ; 0 < rk < rk-1
rk-1 = rkqk+1 + 0
—´Šœ´Êœ rk Ž¸ŠÉ Áž}œÁ«¬˜ª´ ­»—šµo ¥šÉ¸Å¤Än «n œ¼ ¥r ‹³Ážœ} ®.¦.¤. …°Š a ¨³ b

3.4 ‹ÎµœªœÁŒ¡µ³­´¤¡š´ ›r

šœ·¥µ¤ ‹µÎ œªœÁ˜¤È a ¨³ b Ážœ} ‹µÎ œªœÁŒ¡µ³­¤´ ¡š´ ›r „˜È n°Á¤º°É (a, b) = 1

š§¬‘¸šš¸É 8 a ¨³ b Ážœ} ‹ÎµœªœÁŒ¡µ³­´¤¡´š›r „Șn°Á¤°Éº ¤‹¸ µÎ œªœÁ˜¤È x
¨³ y šÉ¸šÎµÄ®o ax + by = 1

š§¬‘¸ ššÉ¸ 9 „µÎ ®œ—‹µÎ œªœÁ˜È¤ a, b ¨³‹µÎ œªœÁŒ¡µ³ p
™µo p | ab ‹³Å—o p | a ¨³ p | b

118 ⌫ ⌫  ⌦
 ⌫     ⌫  

4. „¦³ªœ„µ¦‹—´ „µ¦Á¦¸¥œ¦¼o

4.1 ‡¦¼ššªœ‡ªµ¤¦o¼Á—·¤…°Šœ´„Á¦¸¥œÁ„¸É¥ª„´‡ªµ¤®¤µ¥…°Š˜´ª®µ¦¦nª¤ ¨³˜´ª®µ¦¦nª¤¤µ„

ץĮœo „´ Á¦¸¥œ®µ ®.¦.¤. …°Š‹ÎµœªœšÉ¸„ε®œ— Ánœ

®.¦.¤. …°Š 24 ¨³ 36 (®.¦.¤. ‡°º 12)

®.¦.¤. …°Š 80 ¨³ 56 (®.¦.¤. ‡°º 8)

®.¦.¤. …°Š 0 ¨³ 15 (®.¦.¤. ‡°º 15)

®.¦.¤. …°Š 0 ¨³ 0 (®µ ®.¦.¤. ŤÅn —o)

×¥„µ¦™µ¤˜° ‡¦Â¼ ¨³œ„´ Á¦¸¥œªn ¥„´œ­¦ž» Ážœ} œ¥· µ¤…°Š ®.¦.¤. ŗo—´Šœ¸Ê

šœ¥· µ¤ Ä®o a1, a2, …, an Áž}œ‹ÎµœªœÁ˜È¤ª„šÅ¸É ¤Án žœ} «œ¼ ¥¡r ¦°o ¤„´œ
‹ÎµœªœÁ˜È¤ª„ D š¸É¤¸‡nµ¤µ„š­¸É —» ŽÉŠ¹ D | a1 , D | a2 , … , D | an Á¦¸¥„ªnµÁžœ}
˜´ª®µ¦¦ªn ¤¤µ„ (®.¦.¤.) …°Š a1, a2, …, an
čo­´¨´„¬–r ( a1, a2, …, an ) šœ ®.¦.¤. …°Š a1, a2, …, an

—´ŠœÊ´œ (24, 36) = 12

(80, 56) = 8

( 0, 15) = 15

(0 , 0) ®µÅ¤nŗo
4.2 ‡¦¼°„œ´„Á¦¸¥œªnµ Ĝ„µ¦®µ ®.¦.¤. …°Š‹Îµœªœš¸É¤¸‡nµ¤µ„ Ç Á¦µ­µ¤µ¦™®µÅ—oץčo
…Êœ´ ˜°œ„µ¦®µ¦ ŽŠÉ¹ ¤¸ ɺ°ÁŒ¡µ³ªnµ …Ê´œ˜°œª·›¸¥‡» ¨—· ¨oª‡¦¼¥„˜ª´ °¥nµŠª·›¸®µ ®.¦.¤. —Š´ ˜´ª°¥µn Š˜n°ÅžœÊ¸

˜´ª°¥nµŠ ‹Š®µ ®.¦.¤. …°Š 280 ¨³ 72
ª›· š¸ µÎ Ä®o 72 Áž}œ˜ª´ ®µ¦ —´ŠœÊ¸

280 = 72(3) + 64

72 = 64(1) + 8

…o°­´ŠÁ„˜ 64 = 8(8) + 0
˜ª´ ®µ¦˜ª´ ­—» šoµ¥ šÉ¸šÎµÄ®o Á«¬ = 0 ‹³Ážœ} ®.¦.¤.
—´Šœœ´Ê (280, 72) = 8

(72, 64) = 8

(64, 8) = 8 ŽÉ¹Š­µ¤µ¦™

—Š´ œÊœ´ ™µo m = nq + r ‹³Å—o (m, n) = (n, r) Á­¤°
4.3 ‡¦¼Á­œ°ª·›¸Äœ„µ¦®µ ®.¦.¤. …°Š‹Îµœªœš¸É¤¸‡nµ¤µ„ Ç ˜µ¤…Ê´œ˜°œª·›¸¥»‡¨·—
Á…¸¥œÁžœ} ¦¼ž˜µ¦µŠÅ—o—Š´ œ¸Ê

⌦ 119
⌦

3 280 72 1
216 64 ®.¦.¤.

8 64 8
64
0

¨³‹µ„„µ¦®µ ®.¦.¤. …°Š 280 ¨³ 72 ˜µ¤…´Êœ˜°œª·›¸¥»‡¨·— ­µ¤µ¦™Á…¸¥œ ®.¦.¤. Ĝ¦¼ž
Ÿ¨¦ª¤ÁŠ· Á­œo „¨nµª‡°º Á…¸¥œ 8 Ĝ¦¼žŸ¨¦ª¤Á·ŠÁ­œo …°Š 280 ¨³ 72 ŗo—´ŠœÊ¸

280 = 72(3) + 64 …..(1)

72 = 64(1) + 8 …..(2)

‹µ„ (2) 8 = 72 – 64(1)

8 = 72 – (280 – 72(3))(1)

8 = 72 – 280 + 72(3)

8 = 72(4) + 280(-1)
œœÉ´ ‡º° (280, 72) = 8 = 72(4) + 280(-1)

4.4 Ä®oœ´„Á¦¸¥œšÎµÁ°„­µ¦ f„®´—š¸É 16 …o° 1 Á¡ºÉ°˜¦ª‹­°‡ªµ¤Á…oµÄ‹ ­nªœ…o° 2,3 šÎµÁž}œ
„µ¦oµœ ¡¦o°¤—ªo ¥Â f„®—´ 3.3 „ …°o 1 Ĝ®œŠ´ ­°º Á¦¥¸ œ­µ¦³„µ¦Á¦¥¸ œ¦o¼Á¡¤·É Á˜·¤²

4.5 ÁŒ¨¥Â f„®´—„µ¦oµœ…o°š¸Éœ´„Á¦¸¥œ­Š­´¥ ¨oª‡¦¼„¨nµª™¹Š„µ¦®µ ®.¦.¤. …°Š‹Îµœªœ
®¨µ¥‹Îµœªœ —´Šœ·¥µ¤

šœ¥· µ¤ Ä®o a1, a2, …, an Áž}œ‹ÎµœªœÁ˜È¤ª„šÉŸ ¤Án ž}œ«¼œ¥¡r ¦o°¤„œ´
‹µÎ œªœÁ˜¤È ª„ D š¤É¸ ‡¸ nµ¤µ„šÉ­¸ »— ŽÉŠ¹ D | a1 , D | a2 , … , D | an Á¦¥¸ „ªnµÁžœ}
˜ª´ ®µ¦¦ªn ¤¤µ„ (®.¦.¤.) …°Š a1, a2, …, an
č­o ´¨„´ ¬–r ( a1, a2, …, an ) šœ ®.¦.¤. …°Š a1, a2, …, an

‹µ„šœ¥· µ¤­µ¤µ¦™˜¦ª‹­°Å—ªo nµ (a1, a2, …,an-1,an ) = (a1, a2, …, (an-1, an))

Ä®oœ´„Á¦¸¥œ«¹„¬µ ˜´ª°¥nµŠÃ‹š¥rž{®µ‹µ„®œ´Š­º°Á¦¸¥œ­µ¦³„µ¦Á¦¸¥œ¦¼oÁ¡É·¤Á˜·¤² ®œoµ 107
¨oªÄ®ošÎµÂ f„®´— 3.3 „ …o° 2 – 8 ‹µ„®œ´Š­º°Á¦¸¥œ­µ¦³„µ¦Á¦¸¥œ¦¼oÁ¡·É¤Á˜·¤² ™oµÅ¤nÁ­¦È‹Ä®ošÎµ˜n°
Ážœ} „µ¦oµœ

4.6 ‡¦¼ÁŒ¨¥Â f„®´— 3.3 „ µŠ…o°š¸Éœ´„Á¦¸¥œ­nªœÄ®n¤¸ž{®µ Á¤ºÉ°®¤—…o°­Š­´¥ ‡¦¼
„¨µn ª™Š¹ œ¥· µ¤Â¨³š§¬‘¸šš¸ÉÁ„¸¥É ª…o°Š„´ ®.¦.¤. ץĮoœ´„Á¦¸¥œ¥„˜´ª°¥nµŠ ‹Îµœªœš¸É¤¸ ®.¦.¤. Áž}œ 1
Ánœ

(œ´„Á¦¥¸ œ°µ‹‹³˜°ªnµ (5, 8) = 1

(10, 11) = 1
²¨²)

120 ⌫ ⌫  ⌦
 ⌫     ⌫  

‡¦‹¼ ³°„ªnµ Á¦µ‹³Á¦¥¸ „ ‹ÎµœªœšÉ¤¸ ¸ ®.¦.¤. Ážœ} 1 ªµn ‹µÎ œªœÁŒ¡µ³­´¤¡´š›r Ž¹ÉŠ­¦»žÁž}œœ·¥µ¤Å—o
—Š´ œÊ¸

šœ¥· µ¤ ‹µÎ œªœÁ˜¤È a ¨³ b Áž}œ‹ÎµœªœÁŒ¡µ³­´¤¡š´ ›r „˜È n°Á¤ºÉ° (a, b) = 1

š§¬‘¸š 8 a ¨³ b Áž}œ‹ÎµœªœÁŒ¡µ³­¤´ ¡´š›r „Șn°Á¤ºÉ° ¤¸‹µÎ œªœÁ˜¤È x
¨³ y šš¸É εĮo ax + by = 1

š§¬‘¸ š 9 „µÎ ®œ—‹µÎ œªœÁ˜¤È a, b ¨³‹ÎµœªœÁŒ¡µ³ p
™µo p | ab ‹³Å—o p | a ¨³ p | b

Ä®oœ´„Á¦¸¥œ«¹„¬µ„µ¦¡·­¼‹œr‹µ„®œ´Š­º°Á¦¸¥œ² ®œoµ 116 ‡¦¼°›·µ¥Á¡É·¤Á˜·¤ ×¥
¥„˜ª´ °¥µn Š˜n°Åžœ¸Ê

˜ª´ °¥nµŠš¸É 1 ‹ŠÂ­—Šªµn (a, a+1) = 1
ª·›¸šµÎ Ä®o d = (a, a+1) Á¤Éº° a, k  I
—Š´ œ´Êœ d | a ¨³ d | (a+1)
¨³‹³Å—o d | (( a+1)– a)
σϫ ༡ d | 1

Á¤°Éº d Áž}œ ®.¦.¤. d ˜o°ŠÁžœ} ‹µÎ œªœÁ˜¤È ª„š¸¤É ‡¸ nµ¤µ„šÉ¸­»—
—´Šœ´Êœ d = 1

œœÉ´ ‡º° (a, a+1) = 1

˜ª´ °¥µn ŠšÉ¸ 2 ‹ŠÂ­—Šªnµ a | mn ¨³ (a, m) = 1 ‹³Å—o a | n
ª·›š¸ µÎ Ä®o a | mn
—´Šœœ´Ê ¤¸ x  I ŽŠ¹É mn = ax
¨³Áœ°Éº Š‹µ„ (a, m) = 1
—Š´ œÊœ´ 1 = ap + mq Á¤°Éº p , q  I
‹³Å—o n = nap + nmq
= nap + axq

= a(np + xq)
Áœ°Éº Š‹µ„ (np + xp)  I —´Šœ´œÊ a | n
4.7 Ä®oœ„´ Á¦¥¸ œšÎµÂ f„®—´ 3.3 … …o° 2, 3, 4, 6 Ĝ®œ´Š­°º Á¦¥¸ œ­µ¦³„µ¦Á¦¸¥œ¦Áo¼ ¡É·¤Á˜¤· ²

⌦ 121
⌦

5. ®¨Šn „µ¦Á¦¥¸ œ¦¼o
5.1 Á°„­µ¦ „f ®—´ šÉ¸ 16
5.2 ®œŠ´ ­º°Á¦¥¸ œ­µ¦³„µ¦Á¦¸¥œ¦oÁ¼ ¡·¤É Á˜¤· ²

6. „¦³ªœ„µ¦ª—´ ¨³ž¦³Á¤œ· Ÿ¨

„µ¦ª´—Ÿ¨ „µ¦ž¦³Á¤·œŸ¨
1. ­´ŠÁ„˜‹µ„„µ¦˜°‡Îµ™µ¤ 1. œ„´ Á¦¥¸ œ˜°‡µÎ ™µ¤Å—™o ¼„˜°o ŠÁž}œ­nªœ¤µ„
2. ­´ŠÁ„˜‹µ„‡ªµ¤­œÄ‹ 2. œ„´ Á¦¥¸ œ­œÄ‹Â¨³˜Š´Ê ċÁ¦¸¥œ
3. šµÎ ˚¥rÁ°„­µ¦ f„®—´ šÉ¸ 16 3. œ´„Á¦¸¥œšµÎ ŗ™o „¼ ˜°o Šž¦³¤µ– 80 %
5. šµÎ ˚¥Âr  „f ®—´ 3.3 „ ¨³ … 4. œ´„Á¦¥¸ œšµÎ ŗ™o „¼ ˜°o Šž¦³¤µ– 80 %

Ĝ®œ´Š­°º Á¦¥¸ œ­µ¦³„µ¦Á¦¥¸ œ¦Áo¼ ¡¤·É Á˜¤· ²

7. ´œš„¹ ®¨Š´ „µ¦­°œ
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8. „·‹„¦¦¤Á­œ°Âœ³
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122 ⌫ ⌫  ⌦
 ⌫     ⌫  

Á°„­µ¦ „f ®´—šÉ¸ 16

1. „µÎ ®œ— d = ( 30, 42 ) ‹ŠÁ…¸¥œ d Ĝ¦¼žŸ¨¦ª¤Á·ŠÁ­oœ…°Š 30 ¨³ 42
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2. „ε®œ— d = ( 221, 51 ) ‹ŠÁ…¸¥œ d Ĝ¦¼žŸ¨¦ª¤Á·ŠÁ­oœ…°Š 221 ¨³ 51
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3. „µÎ ®œ— d = ( 147, 56 ) ‹ŠÁ…¥¸ œ d Ĝ¦¼žŸ¨¦ª¤ÁŠ· Á­œo …°Š 147 ¨³ 56
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4. ‹Š®µ‹µÎ œªœÁ˜¤È x ¨³ y šÉ¸­°—‡¨o°Š„´­¤„µ¦ ( 144, 308 ) = 144x + 308y
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⌦ 123
⌦

Ÿœ„µ¦‹´—„µ¦Á¦¸¥œ¦š¼o ɸ 17

Á¦ºÉ°Š ˜´ª‡¼–¦ªn ¤œo°¥ (‡.¦.œ.) ´œÊ ¤´›¥¤«„¹ ¬µžše ¸É 4
ª·µ ‡–˜· «µ­˜¦r Áª¨µ 2 ´ÉªÃ¤Š

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Ÿ¨„µ¦Á¦¥¸ œ¦š¼o ¸É‡µ—®ª´Š
°„‡ªµ¤®¤µ¥ č­o ´ ¨„´ ¬–…r °Š‡.¦.œ. ¨³œÎµ‡ªµ¤¦Á¼o ¦ºÉ°Š ‡.¦.œ. ŞčÅo —o

1. ‹—» ž¦³­Š‡r„µ¦Á¦¸¥œ¦o¼
1.1 °„‡ªµ¤®¤µ¥ ¨³ ®µ ‡.¦.œ.ŗo
1.2 °„‡ªµ¤­´¤¡œ´ ›¦r ³®ªnµŠ ®.¦.¤. ¨³ ‡.¦.œ. ŗo
1.3 œµÎ ‡ªµ¤¦¼Áo ¦É°º Š ‡.¦.œ. Şčoŗo

2. œª‡ªµ¤‡—· ®¨„´
‹ÎµœªœÁ˜È¤ª„ c šÉ¸¤¸‡nµœo°¥š¸É­»—‹³Áž}œ˜´ª‡¼–¦nª¤œo°¥ (‡.¦.œ.) …°Š‹ÎµœªœÁ˜È¤ a ¨³ b

„Șn°Á¤º°É a | c ¨³ b | c č­o ´ ¨„´ ¬–r [ a , b ] šœ ‡.¦.œ. …°Š a ¨³ b

3. Áœº°Ê ®µ­µ¦³

šœ·¥µ¤ Ä®o a ¨³ b Áž}œ‹ÎµœªœÁ˜¤È š¸ÉŤnÁžœ} «¼œ¥r
‹µÎ œªœÁ˜È¤ª„ c šÉ¤¸ ‡¸ nµœo°¥š¸­É »— Ž¹ŠÉ a | c ¨³ b | c Á¦¥¸ „ªnµ
˜ª´ ‡¼–¦nª¤œo°¥ (‡.¦.œ.) …°Š a ¨³ b
čo­´¨„´ ¬–r [ a , b ] šœ ‡.¦.œ. …°Š a ¨³ b

šœ·¥µ¤ Ä®o a1 , a2 ,…,an Áž}œ‹µÎ œªœÁ˜¤È šÉŸ ¤nÁž}œ«œ¼ ¥r
‹ÎµœªœÁ˜È¤ª„ C š¸É¤¸‡µn œo°¥š­É¸ »— Ž¹ÉŠ a1 | C , a2 | C , … , an | C Á¦¸¥„ªnµ
˜ª´ ‡–¼ ¦ªn ¤œo°¥ (‡.¦.œ.) …°Š a1 , a2 ,…,an
čo­´ ¨„´ ¬–r [a1 , a2 ,…,an ] šœ ‡.¦.œ. …°Š a1 , a2 ,…,an

š§¬‘¸ š 10 ™µo a ¨³ b Ážœ} ‹µÎ œªœÁ˜¤È ª„ ¨oª ab = (a , b) [a , b]

4. „¦³ªœ„µ¦‹—´ „µ¦Á¦¸¥œ¦¼o
4.1 ‡¦¼ššªœ„µ¦®µ ‡.¦.œ. ×¥„ε®œ—‹Îµœªœ­°Š‹Îµœªœ ¨³Ä®oœ„´ Á¦¸¥œªn ¥„´œ®µ ‡.¦.œ.

Áœn Ä®o®µ ‡.¦.œ. …°Š 36 ¨³ 24 (Ž¹ÉŠœ„´ Á¦¥¸ œ°µ‹‹³®µÃ—¥ª›· ¸ ¥„˜ª´ ž¦³„°®¦º°®µ¡®‡» –¼ ¦nª¤)
4.2 ‡¦Ä¼ ®œo ¥· µ¤ ‡.¦.œ.

124 ⌫ ⌫  ⌦
 ⌫     ⌫  

šœ¥· µ¤ Ä®o a1 , a2 ,…,an Ážœ} ‹µÎ œªœÁ˜È¤šÉŸ ¤nÁž}œ«œ¼ ¥r ‹ÎµœªœÁ˜¤È ª„ C šÉ¸¤¸‡µn œ°o ¥
šÉ­¸ —» ŽŠ¹É a1 | C , a2 | C , … , an | C Á¦¥¸ „ªnµ ˜ª´ ‡¼–¦nª¤œo°¥ (‡.¦.œ.) …°Š a1 , a2 ,…,an

č­o ´ ¨´„¬–r [a1 , a2 ,…,an ] šœ ‡.¦.œ. …°Š a1 , a2 ,…,an
4.3 œ´„Á¦¸¥œšÎµÂ f„®´—šÉ¸ 3.4 …o° 1 Á­¦È‹Â¨oªÄ®oœ´„Á¦¸¥œ«¹„¬µ˜´ª°¥nµŠšÉ¸ 4 – 5 Ĝ®œ´Š­º°

Á¦¥¸ œ ‡¦°¼ ›· µ¥Á¡·É¤Á˜¤·

4.4 Ä®oœ´„Á¦¸¥œ¥„˜´ª°¥nµŠ‹ÎµœªœÁ˜È¤ª„¤µ‡œ¨³­°Š‹Îµœªœ ¨³Ä®oœ´„Á¦¸¥œ®µ ‡.¦.œ. ¨³

®.¦.¤. ¨³Ä®oœ´„Á¦¸¥œ˜¦ª‹­°ªnµ ‡.¦.œ. x ®.¦.¤. = Ÿ¨‡¼–…°ŠÁ¨…­°Š‹ÎµœªœšÉ¸œ´„Á¦¸¥œ®µ¤µ

®¦°º Ťn

4.5 ‡¦¼Ä®šo §¬‘¸ : ™oµ a , b Áž}œ‹ÎµœªœÁ˜¤È ª„¨ªo

ab = (a , b) [a , b] (Ä®œo ´„Á¦¥¸ œÅž¡·­‹¼ œrÁžœ} „µ¦oµœ)

4.6 ‡¦¼Ä®o˜´ª°¥nµŠ ™oµ‹Îµœªœ x ¨³ 30 ¤¸ ®.¦.¤. Áž}œ 6 ¨³ ‡.¦.œ. Áž}œ 125 ‹Š®µªnµ x

Ážœ} ‹µÎ œªœÄ—

ª›· ¸šµÎ (x , 30) [x , 30] = x u 30

(6) (125) = 30x

25 = x

4.7 ‡¦¼Ä®oœ´„Á¦¸¥œšÎµÂ f„®´—šÉ¸ 3.4 ®œoµ 113 …o° 2 , 3 , 4 , 5 Ĝ®œ´Š­º°­µ¦³„µ¦Á¦¸¥œ¦o¼
Á¡É¤· Á˜·¤ ²

5. ®¨nŠ„µ¦Á¦¸¥œ¦o¼
®œŠ´ ­°º ­µ¦³„µ¦Á¦¥¸ œ¦oÁ¼ ¡¤É· Á˜¤· ‡–·˜«µ­˜¦r Á¨¤n 1 œ´Ê ¤´›¥¤«„¹ ¬µžše ¸É 4

6. „¦³ªœ„µ¦ª—´ ¨³ž¦³Á¤œ· Ÿ¨ „µ¦ž¦³Á¤·œŸ¨

„µ¦ª—´ Ÿ¨ 1. œ„´ Á¦¥¸ œ˜°‡µÎ ™µ¤Å—™o „¼ ˜°o ŠÁž}œ­nªœ¤µ„
1. ­´ŠÁ„˜‹µ„„µ¦˜°‡Îµ™µ¤ 2. œ„´ Á¦¸¥œ­œÄ‹Â¨³˜ÊŠ´ ċÁ¦¥¸ œ
2. ­Š´ Á„˜‹µ„‡ªµ¤­œÄ‹ 3. œ„´ Á¦¥¸ œšµÎ ŗ™o ¼„˜°o Šž¦³¤µ– 80 %
3. šµÎ ˚¥rÁ°„­µ¦ f„®—´ 3.4 ®œµo 113

7. œ´ š¹„®¨´Š­°œ
……………………………………………………………………………………………………………
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8. „·‹„¦¦¤Á­œ°Âœ³
……………………………………………………………………………………………………………
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⌦ 125
⌦

 „f ®´—¦³‡œ

‹ŠÁ¨º°„‡Îµ˜°š¸É™„¼ ˜°o Š

1. Ä®o x, y ¨³ z Áž}œ‹Îµœªœ‹¦·Š ¨³ x ' y = 3x2 y ‡nµ…°Š (z ' x) 'y ‡º°…°o ė

1. 9 z3x2y 2. 3 x2yz

3. 27 x2yz4 4. 9 x4y2z

2. …o°‡ªµ¤Ä—˜°n Şœ¸Ê™¼„˜o°Š
1. ™oµ x > y ¨oª 1 < 1

xy

2. Ÿ¨˜µn Š…°Š‹Îµœªœ°˜¦¦„¥³„´‹Îµœªœ°˜¦¦„¥³ ¥n°¤Ážœ} ‹Îµœªœ°˜¦¦„¥³
3. ™µo x < 0 ¨oª x2 = -x
4. °œ· Áª°¦r­­Îµ®¦´„µ¦‡–¼ …°Š 3 + 5 ‡º° 3 - 5

3. Ä®o x y < 5 ¨³ x z < 4 ˜°n Şœ…¸Ê °o ė™„¼ ˜°o Š

1. x2 xz xy yz < 20 x y 5

2. <

x z 4

3. y z < 1 4. 2x y z < 19

4. …°o ‡ªµ¤Ä—˜n°ÅžœÅʸ ¤‹n ¦Š·

1. ™oµ x Áž}œ‹Îµœªœ˜¦¦„¥³Â¨oª ‹³Å¤n­µ¤µ¦™®µ x ŽÉŠ¹ ¤‡¸ nµœo°¥š­É¸ —» ×¥š¸É x < 9

2. ™µo a Ážœ} ‹ÎµœªœÁ˜È¤šÉ¸Å¤Án ž}œ«œ¼ ¥r¨ªo ‹³¤¸‹µÎ œªœÁ˜¤È p ¨³ q ŽÉŠ¹ p z a q z 0

¨³ p

=a
q

3. ™µo a Ážœ} ‹ÎµœªœÁ˜È¤š¸ÉŤnÁžœ} ‹Îµœªœ˜¦¦„¥³Â¨oª ‹³Á…¥¸ œ a ŗoĜ¦¼žš«œ¥· ¤Å¤Žn Êε

4. ™µo a Áž}œ‹µÎ œªœ‹¦Š· ¨oª n an = a Á¤ºÉ° n = 2 4 6 …

5. Ä®o A = ^x _ x = 3n n Áž}œ‹ÎµœªœÁ˜¤È ª„` …°o ‡ªµ¤Ä—˜n°Åžœ™Ê¸ ¼„˜o°Š
1. ÁŽ˜ A ¤‡¸ –» ­¤˜´ ž· —d …°Š„µ¦ª„
2. ÁŽ˜ A ¤‡¸ –» ­¤´˜ž· —d …°Š„µ¦‡¼–
3. 0 Ážœ} Á°„¨„´ ¬–r„µ¦ª„ÄœÁŽ˜ A
4. 1 Ážœ} Á°„¨„´ ¬–r„µ¦‡–¼ ĜÁŽ˜ A

6. „µÎ ®œ—Ä®o xy t 0 ×¥šÉ¸ x y Ážœ} ‹µÎ œªœ‹¦·Š …°o ė˜°n Şœ™Ê¸ ¼„˜°o Š

1. x + y t 0 2. x + y t 0

3. x y < x + y 4. x y = x + y

126 ⌫ ⌫  ⌦
 ⌫     ⌫  

7. Ä®o a b c ¨³ d Ážœ} ‹Îµœªœ‹¦Š· ‹Š¡‹· µ¦–µªnµ…o°Ä—˜°n Şœ™Ê¸ „¼ ˜°o Š

1. ab > 0 „Șn°Á¤ºÉ° a > 0 ¨³ b > 0 Ášµn œ´œÊ

2. ™oµ b < a ¨ªo 11

<
ab

3. ™µo a Áž}œ‹µÎ œªœ°˜¦¦„¥³Â¨ªo a Áž}œ‹Îµœªœ°˜¦¦„¥³

4. ™oµ 0 < a < 1 ¨oª a2 > a

8. …o°‡ªµ¤˜n°Åžœ¸Ê …°o ėŸ—·

1. ™oµ a > 0 b > 0 ¨³ a z b ¨oª a b < 2 Á­¤°

ba

2. ™oµ a>0 b>0 ¨³ a z b ¨ªo a b > 1 1 Á­¤°
b2 a2 ab

3. ™oµ a2 + b2 =1 ¨³ c2 + d2 =1 ¨ªo ac + bd d 1 Á­¤°

4. ™oµ a ¨³ b Ážœ} ‹Îµœªœ‹¦·ŠÂ¨oª ax + b = 0 Ťn‹µÎ Ážœ} ˜°o Š¤Á¸ ¡¸¥Š‡Îµ˜°Á—¥¸ ª

9. ™oµ x ¨³ y Áž}œ‹µÎ œªœ‹¦·ŠÄ—Ç ‹Š¡·‹µ¦–µ…°o ‡ªµ¤Ä—˜n°Åžœ¸Ê

„. ™µo xy = 0 ¨oª x Ťn¤¸‡ªµ¤®¤µ¥

y

…. ™µo xy = 0 ¨oª y Ťn¤‡¸ ªµ¤®¤µ¥

x

‡. xy = a „˜È n°Á¤ºÉ° x = a

y

…o°Ä—˜°n ޜʸ™„¼ ˜°o Š

1. …°o „. ¨³ …. Ášnµœœ´Ê šÉ™¸ ¼„˜°o Š 2. …o° „. ¨³ ‡. Ášnµœ´Êœš™¸É „¼ ˜°o Š

3. …°o …. ¨³ ‡. Ášµn œÊœ´ šÉ™¸ ¼„˜°o Š 4. …°o „. …. ¨³ ‡. Ÿ·—®¤—

10. Ä®o R šœÁŽ˜…°Š‹µÎ œªœ‹¦·Š …o°Ä—˜°n Şœ¸Ê™„¼ ˜°o Š
1. ™oµ a ¨³ b Áž}œ‹µÎ œªœ˜¦¦„¥³ ¨³ c Ážœ} ‹µÎ œªœ°˜¦¦„¥³ ¨ªo a + bc Ážœ}
‹µÎ œªœ°˜¦¦„¥³

2. ™µo a b  R ¨³ a > b ¨ªo a b - a > b - a b
3. ™µo a b c  R ¨³ a b < 0 < c ¨ªo c c

ab

4. ™oµ a b c d  R ¨³ 0 < a b ¨³ c d < 0 ¨oª a d 2 b c2

11. ÁŽ˜‡Îµ˜°…°Š°­¤„µ¦ 1 2 Ášµn „´ …o°Ä—

x 1 3x 1

1. -f -1 2. -f -3

3. 1 3 4. -f -1 ‰ 1 3

3 3

⌦ 127
⌦

12. Ä®o x Áž}œ‹µÎ œªœ‹¦Š· Ä—Ç ‹Š¡·‹µ¦–µ…o°‡ªµ¤Ä—˜°n Şœ¸Ê

„. ™oµ A Ážœ} ÁŽ˜‡Îµ˜°…°Š 2 3x 2 3 x ¨³ B Ážœ} ÁŽ˜‡Îµ˜°…°Š

2 3x 2 3x ¨oª A Áž}œ­´ ÁŽ˜Âšo…°Š B

…. A = 1 3 ¨³ B Áž}œÁŽ˜‡Îµ˜°…°Š x 3 d 0

3 3x 1

¨ªo A ˆ B = A

˜°n Şœ¸Ê…°o ė™„¼ ˜o°Š

1. „. ¨³ …. ™„¼ š´ÊŠ­°Š…°o 2. ™¼„ÁŒ¡µ³…o° „.

3. ™„¼ ÁŒ¡µ³…o° …. 4. „. ¨³ …. Ÿ·—šŠ´Ê ­°Š…°o

13. Ä®o A Áž}œÁŽ˜‡Îµ˜°…°Š x 2 x d 4 ¨³ B Áž}œÁŽ˜‡Îµ˜°…°Š°­¤„µ¦ x x 7

2

‹Š®µ A ˆ B = A

1. -f 2 2. -f 7

2

3. ‡ 4. -f -2

14. Ä®o x , y , z Áž}œ‹ÎµœªœÁ˜È¤ª„š¸É¤¸‡nµÁ¦¸¥Š˜·—„´œ‹µ„œo°¥Åž®µ¤µ„ ™oµ y Áž}œ‹ÎµœªœÁ˜È¤
ª„šÉ¤¸ ‡¸ nµœo°¥š¸­É »—š¸šÉ εĮo 3 x y z Ážœ} ‹ÎµœªœÁ˜¤È ª„ ¨ªo y ¤‡¸ µn Ášµn ė
(Ent. ‡–·˜ 1 ˜.‡. 2543)

1. 1 2. 3 3. 7 4. 9

15. Ä®o S = { 0 , 1 , 2,…,7 } ¨³œ·¥µ¤ a
b = Á«¬Á®¨º°‹µ„„µ¦®µ¦Ÿ¨‡¼– ab —oª¥ 6

š„» a,b  s ¡‹· µ¦–µ…o°‡ªµ¤˜n°ÅžœÊ¸

„. x
1 = x š»„ x  s …. {4
x _ x  s} = {0, 2 ,4}

…°o ė˜°n Şœ¸ÊÁž}œ‹¦Š· (Ent. ‡–˜· 1 ¤¸.‡. 2543)

1. „. ™„¼ ¨³ …. ™¼„ 2. „. ™¼„ ˜n …. Ÿ—·

3. „. Ÿ·— ˜n …. ™¼„ 4. „. Ÿ·— ¨³ …. Ÿ—·

16. „ε®œ—Ä®o x + 1 ¨³ x – 1 Áž}œ˜´ªž¦³„°…°Š¡®»œµ¤ p(x) = 3x3 + x2 – ax + b Á¤Éº° a , b
Áž}œ‡µn ‡Š˜ª´ Á«¬Á®¨°º š¸ÅÉ —‹o µ„„µ¦®µ¦ p(x) —oª¥ x - a – b Ášµn „´…o°Ä—˜°n ޜʸ
(Ent. ‡–˜· 1 ¤.¸ ‡. 2544)

1. 15 2. 17 3. 19 4. 21

128 ⌫ ⌫  ⌦
 ⌫     ⌫  

⌦ 129
⌦

130 ⌫ ⌫  ⌦
 ⌫     ⌫  

36. Ä®o x Áž}œ‹ÎµœªœÁ˜È¤ª„š¸É¤¸‡nµ¤µ„šÉ¸­»—šÉ¸®µ¦ 323 , 227 ¨³ 155 ¨oª¤¸Á«¬Á®¨º° r
Ášµn „´œ —Š´ œœÊ´ Á¤°Éº ®µ¦ x —oª¥ r ‹³¤Á¸ «¬Á®¨°º Ášµn „´ Ášµn ė
1. 1 2. 2 3. 3 4. 4

⌦ 131
⌦

37. ‹ÎµœªœÁ˜È¤˜Ê´ŠÂ˜n 0 ™¹Š 100 šÉ¸Å¤nÁž}œ‹ÎµœªœÁŒ¡µ³­´¤¡´š›r „´ 15 ¤¸š´ÊŠ®¤—„¸É‹Îµœªœ
(Ent. ‡–˜· «µ­˜¦r „… že 2537)

1. 48 2. 47 3. 46 4. 45

38. Ĝ„µ¦œµÎ …´œÊ ˜°œª·›„¸ µ¦®µ¦ÅžÄÄo œ„µ¦Á…¸¥œ‹µÎ œªœ 118 Ĝ¦ž¼ ˜ª´ Á¨…“µœ 4
‹³Å—o 118 = (4 u a) + b
a = (4 u c) + d
c = (4 u e) + f
e = (4 u 0) + e
—Š´ œ´Êœ bdef Ášnµ„´ÁšnµÄ—

1. 6 2. 8 3. 12 4. 16

39. „µÎ ®œ— A = {x  I+~(3x + 2 , 5x + 3) = 2}
B = {x  I+~x Áž}œ¡®‡» ¼–…°Š 24 , 100 d x d 300 ¨³ 5~x}

—Š´ œ´œÊ B – A Ážœ} ­´ÁŽ˜…°ŠÁŽ˜Ä—

1. {100 , 120 , 140 , 160} 2. {200 , 240 , 280 , 320}

3. {240 , 480 , 720 , 960} 4. {120 , 240 , 360 , 480}

40. „ε®œ— A = {x  I+~x Áž}œ‹ÎµœªœÁ˜È¤ª„šÊ´Š®¤—šÉ¸®µ¦ 215 ¨³ 267 ¨oª¤¸Á«¬Á®¨º°
Ášµn „´œ } ‹µÎ œªœ­¤µ·„…°Š A Ášµn „´Ášµn ė

1. 6 2. 7 3. 8 4. 9

41. ‹µÎ œªœÁ˜¤È x ¨³ y ­°—‡¨o°Š„´ ­¤„µ¦ (116 , 248) = 116x + 248y ™µo x
y = xy – (y – x)
¨oª x
y Ášnµ„´ Ášµn ė

1. –137 2. –113 3. –97 4. –83

42. ­Îµ®¦´‹ÎµœªœÁ˜È¤ a , b Ä—Ç Ä®o (a , b) Ášnµ„´ ®.¦.¤. …°Š a ¨³ b Ä®o A = {1,2,3,...,400}
‹Îµœªœ­¤µ·„…°ŠÁŽ˜ {x  A~(x , 40) = 5} ¤¸‡nµÁšnµ„´…o°Ä—˜n°Åžœ¸Ê (Ent. ‡–·˜«µ­˜¦r
1 ˜¨» µ‡¤ 2542)

1. 30 2. 40 3. 60 4. 80

43. „ε®œ—Ä®o S = {n  I+~n d 1000 ,®.¦.¤. …°Š n ¨³ 100 Ášnµ„´ 1} ‹Îµœªœ­¤µ·„…°ŠÁŽ˜
S Ášnµ„´ ÁšnµÄ— (Ent. ‡–·˜«µ­˜¦r 1 ¤¸œµ‡¤ 2545)

1. 600 2. 500 3. 400 4. 300

132 ⌫ ⌫  ⌦
 ⌫     ⌫  

44. ™oµ A = {p~p Áž}œ‹ÎµœªœÁŒ¡µ³ª„ ¨³ p~(980 - p)3} ¨oªŸ¨ª„…°Š­¤µ·„šÊ´Š®¤—
Ĝ A ¤‡¸ nµÁšnµÄ— ( Ent. ‡–˜· «µ­˜¦r 1 ¤œ¸ µ‡¤ 2542)

1. 14 2. 17 3. 21 4. 19

45. „ε®œ—Ä®oÁ°„£¡­´¤¡´š›r‡º° {x~x Áž}œ‹ÎµœªœÁ˜È¤šÉ¸Å¤nčn 0 ¨³ -100 d x d 100} Ä®o
A = {x~®.¦.¤. …°Š x „´ 21 Ážœ} 3} ‹Îµœªœ­¤µ·„…°Š A Ášµn „´ …°o ė˜°n ޜʸ

1. 29 2. 34 3. 58 4. 68

46. „µÎ ®œ— a , b , c Ážœ} ‹µÎ œªœÁ˜¤È ª„ 4. (a , b) = 5
¡‹· µ¦–µ a = bc + r , 0 < r < b

b = r(2) + r1 , 0 < r1 < r

r = r1(3) + r2 , 0 < r2 < r1
™µo r1 = 5 ¨³ r2 = 0 …o°Ä—„¨µn ª™¼„˜o°Š

1. (a , b) = 1 2. (a , b) = 2 3. (a , b) = 3

47. Ä®o a , b Áž}œ‹ÎµœªœÁ˜È¤ª„ŽÉŠ¹ a < b , 5 ®µ¦ a ¨Š˜´ª ¨³ 3 ®µ¦ b ¨Š˜´ª ™oµ a , b Áž}œ‹Îµœªœ
ÁŒ¡µ³­´¤¡´š›r ¨³ ‡.¦.œ. …°Š a , b Ášnµ„´ 65 ¨oª a ®µ¦ b Á®¨º°Á«¬Ášnµ„´…o°Ä—˜n°ÅžœÊ¸
(Ent. ‡–·˜«µ­˜¦r „… Á¤¬µ¥œ 2541)

1. 1 2. 2 3. 3 4. 4

48. Ä®o n Áž}œ‹µÎ œªœÁ˜È¤ª„ŽÉŠ¹ ®.¦.¤.…°Š n ¨³ 42 Ášnµ„´ 6
™oµ 42 = nq0 + r0 0 < r0 < n
n = 2r0 + r1 0 < r1 < r0
¨³ r0 = 2r1
×¥šÉ¸ q0 r0 r1 Áž}œ‹µÎ œªœÁ˜¤È ¨ªo ‡.¦.œ.…°Š n ¨³ 42 ¤‡¸ nµÁšµn „´ Ášµn Ŧ
(Ent. ‡–·˜«µ­˜¦r „… Á¤¬µ¥œ 2541)

1. 180 2. 190 3. 200 4. 210

49. „µÎ ®œ— a , b Ážœ} ‹µÎ œªœÁ˜¤È ™oµ (a , b) = 1 ¨ªo (a+b , a-b) Ášnµ„´ Ášµn Ŧ
1. 1 2. 2
3. 3 4. ¤‡¸ 嘰¤µ„„ªnµ 1 …o°

⌦ 133
⌦

50. Ä®o x ¨³ y Áž}œ‹ÎµœªœÁ˜È¤ª„ ŽÉ¹Š 80 < x < 200 ¨³ x = pq Á¤Éº° p ¨³ q Áž}œ
‹ÎµœªœÁŒ¡µ³ ŽÉ¹Š p z q ™oµ x ¨³ y Áž}œ‹ÎµœªœÁŒ¡µ³­´¤¡´š›r ¨³ ‡.¦.œ. …°Š x , y
Ášnµ„´ 15,015 ¨oªŸ¨ª„…°Š‡nµ…°Š y šÊ´Š®¤—šÉ¸­°—‡¨o°Š„´ÁŠºÉ°œÅ…šÊ´Š®¤—š¸É„ε®œ—Ä®o
Ášµn „´ÁšnµÄ— ( Ent. ‡–˜· «µ­˜¦r „… že 2538)

1. 250 2. 270 3. 290 4. 310

51. „ε®œ—Ä®o x ¨³ y Áž}œ‹ÎµœªœÁ˜È¤ª„ ×¥š¸É x < y ®.¦.¤. …°Š x , y Ášnµ„´ 9 ‡.¦.œ. …°Š
x , y Ášnµ„´ 28,215 ¨³‹ÎµœªœÁŒ¡µ³šÉ¸Â˜„˜nµŠ„´œšÊ´Š®¤—šÉ¸®µ¦ x ¨Š˜´ª ¤¸ 3 ‹Îµœªœ ‡nµ…°Š
y – x Ášnµ„´…o°Ä—˜°n ޜʸ ( Ent. ‡–·˜«µ­˜¦r „… že 2537)

1. 46 2. 35 3. 18 4. 9

52. „µÎ ®œ— (65 , 78) = m ¨³ [72 , 96] = n ¡·‹µ¦–µ…°o ‡ªµ¤˜°n Şœ¸Ê
„. m ¨³ n Áž}œ‹ÎµœªœÁŒ¡µ³­´¤¡´š›r
…. m /n ×¥¤Á¸ «¬Á®¨º°Ážœ} 2
‡. n = 192m

…°o ‡ªµ¤Ä—„¨nµª™„¼ ˜o°Š
1. „ ¨³ … 2. „ ¨³ ‡ 3. … ¨³ ‡ 4. „ , … ¨³ ‡

53. „µÎ ®œ— a , b Ážœ} ‹µÎ œªœÁ˜¤È ×¥š¸É 0 < a < b ¨³ (a , b)[a , b] = 15 ¡‹· µ¦–µ

(1) a + b œo°¥š¸­É —» Ášnµ„´ 8 (2) ™oµ [a + 1 , b + 1] ¤µ„š¸É­»—Ášµn „´ 32

…o°Ä—„¨µn ª™¼„˜°o Š

1. …o° (1) ™¼„ ¨³…o° (2) ™¼„ 2. …°o (1) ™¼„ ¨³…o° (2) Ÿ—·

3. …°o (1) Ÿ·— ¨³…°o (2) ™¼„ 4. …o° (1) Ÿ—· ¨³…o° (2) Ÿ—·

***********************************************************************************

134 ⌫ ⌫  ⌦
 ⌫     ⌫  

šž¸É ¦„¹ ¬µ : Ÿ—o¼ µÎ Áœ·œ„µ¦
—¦.°µÎ ¦Š» ‹´œšªµœ·
—¦.­¦· ¡· ¦ » µœ´œ˜r Á¨…µ›„· µ¦­£µ„µ¦«„¹ ¬µ
¦«.—¦.­Îµ°µŠ ®¦· ´¦¼ –³ ¦°ŠÁ¨…µ›·„µ¦­£µ„µ¦«¹„¬µ
—¦.¦nŠ» Á¦º°Š ­»…µ£·¦¤¥r …µo ¦µ„µ¦µÎ œµ š¸žÉ ¦„¹ ¬µÃ‡¦Š„µ¦²
œµŠ­µª­»š›µ­œ· ¸ ª´ ¦¼¨ Ÿo¼˜¦ª‹¦µ„µ¦„¦³š¦ªŠ«„¹ ¬µ›·„µ¦š¸Éž¦¹„¬µÃ‡¦Š„µ¦²
—¦.‹·¦¡¦¦– ž–» Á„¬¤ šÉ¸ž¦„¹ ¬µ—µo œ¦³„µ¦«„¹ ¬µ ­„«.
Ÿ°o¼ µÎ œª¥„µ¦­µÎ œ´„¤µ˜¦“µœ„µ¦«„¹ ¬µÂ¨³¡´•œµ„µ¦Á¦¥¸ œ¦¼o

Ÿ¼Áo ¦¥¸ Á¦¸¥Š : œµŠ‹¦¦¥µ ¡œ´ ›rÁ» ™¨Š· °¤¦ æŠÁ¦¸¥œ®µ—Ä®nªš· ¥µ¨¥´ ‹Š´ ®ª—´ ­Š…¨µ

Ÿo˜¼ ¦ª‹šµœ :

¦°Š«µ­˜¦µ‹µ¦¥r°µ¦­· µ ¦˜´ œÁ¡È¦r ®ª´ ®œµo ‡–³ª‹· ´¥

—¦.«£» ª¦¦– Á¨«· ń¦

°µ‹µ¦¥rÁ°›­´ ª•´ œr ‡Îµ¤–¸

°µ‹µ¦¥­r »›·˜µ ¤–¸ ´¥

‡–³°µ‹µ¦¥Ÿr ¼­o °œ‡–·˜«µ­˜¦Ãr ¦ŠÁ¦¸¥œš¸ÉÁ…oµ¦nª¤Ã‡¦Š„µ¦² ‹µ„æŠÁ¦¥¸ œ—´Š˜n°ÅžœÊ¸

x æŠÁ¦¸¥œ®µ—Ä®ªn ·š¥µ¨¥´ ‹´Š®ª—´ ­Š…¨µ

x æŠÁ¦¥¸ œ¤®µª¦· µª›» ‹Š´ ®ª—´ ­Š…¨µ

x æŠÁ¦¸¥œ¦¼ –³¦µÎ ¨„¹ ‹Š´ ®ª—´ ˜¦Š´

x æŠÁ¦¥¸ œ‹¯» µ£¦–¦µª·š¥µ¨¥´ ‹´Š®ª—´ ­˜¼¨

x æŠÁ¦¸¥œ­»¦µ¬’¦›r µœ¸ ‹´Š®ª—´ ­»¦µ¬’¦›r µœ¸

x æŠÁ¦¥¸ œ¡œ» ¡œ· ¡š· ¥µ‡¤ ‹´Š®ª—´ ­¦» µ¬’¦r›µœ¸

x æŠÁ¦¥¸ œÁ˜¦¥¸ ¤°—» ¤£µ‡Ä˜o ‹Š´ ®ª—´ œ‡¦«¦›¸ ¦¦¤¦µ

Ÿ¼o¡·‹µ¦–µ¦µ¥Šµœ : œµ¥­¤µ¥ «¦ª¸ ¦µŠ„¨¼ æŠÁ¦¸¥œÁ˜¦¥¸ ¤°—» ¤«„¹ ¬µ „¦Š» Áš¡²

Ÿ¼¦o ´Ÿ—· °Ã‡¦Š„µ¦ : ®´ª®œµo ǦŠ„µ¦
œµŠ­µª» Áš¥¸ ¤ «·¦·ž{ µ ž¦³‹ÎµÃ‡¦Š„µ¦
œµ¥¦ª· ˜µÂ„oª ž¦³‹µÎ ǦŠ„µ¦
œµŠ­µª„ŠÉ· „µ‹œr Á¤‰µ ž¦³‹µÎ ǦŠ„µ¦
œµ¥«·¦·¦˜´ œr µÎ œµ„·‹

¦¦–µ›·„µ¦ :
œµŠ­µª»Áš¥¸ ¤ «·¦ž· { µ
œµŠ­µª„ÉŠ· „µ‹œr Á¤‰µ

Á¦¸¥Á¦¸¥ŠÂ¨³‹´—šµÎ ¦µ¥Šµœ :
œµŠ­µª„ŠÉ· „µ‹œr Á¤‰µ

⌦ 135
⌦

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