Chuyen de HSG CM chia het

Chuyªn ®Ò chøng minh chia hÕt

( Mêi c¸c thÇy c« xem vµ cho ý kiÕn do thêi gian chuÈn bÞ cha ®îc kü mong thÇy c« bá qua,

ngêi viÕt NguyÔn Thanh Hïng §VCT Trêng THCS Tiªn Nha –Lôc Nam B¾c Giang §T

0986713720 ®éc gi¶ ®îc chØnh söa tho¶i m¸i)

I. D¹ng bµi chøng minh chia hÕt khi vµ chØ khi.

1) dcba4 ⇔(a +2b)4 11) (3a +2b)17 ⇔(10a +b)17

2) N =dcba8 ⇔(a +2b +4c)8 12) abb7 ⇔(a +2b)7

3) N =dcba16 ⇔(a +2b +4c +8d)16 , b ch½n. 13) abcd101⇔ab =cd

4) ab13 ⇔(a +4b)13 14) (a −5b)17,CMR,(10a +b)17

5) ab17 ⇔(3a +2b)17 15) Chøng minh r»ng ( 3x +5y)17 th×

(x +4y)17 ®iÒu ngîc l¹i cã ®óng

kh«ng x, y ∈N .

6) abcd 29 ⇔(a +3b +9c +27d)29

7) (2x +3y)17 ⇔(9x +5y)17

8) (11a +2b)19 ⇔(18a +5b)19

9) (a +4b)13 ⇔(10a +b)13

10) (abc37,CMR,bca,cab37

II. D¹ng T×m sè sö dông dÊu hiÖu chia hÕt.

Bµi tËp: T×m x, y sao cho:

1) 135x4y45 9) 7xy963

2) 1234xy72 10) 6x5y2;9

3) 34x5y36 11) 75xy45

4) 64x5y36 12) 31x4y2;5;4

5) 71x1y45 13) 17xy2;3 chia cho 5 d 1

6) 56x3y36

7) 135xy45

8) 47x5y28

III. D¹ng bµi chøng minh chia hÕt sö dông ph ¬ng ph¸p ®Æt nh©n tö chung.

1) (120a +36b)12 16) (7 7 7 ) 11 6 5 4

+ − 

2) (5 5 5 ) 21 7 6 5

− +  17) (81 27 9 ) 45 7 9 13

− − 

3) (4 4 4 ) 28 39 46 41 + +  18) (10 10 10 ) 555 9 8 7

+ + 

4) (5 5 5 ) 31 2003 2002 2001 + +  19) (3 3 3 ... 3 ) 13;31 3 5 1991 + + + + 

5) (7 7 7 ) 77 6 5 4

+ −  20) (11 11 11 ... 11 1) 5

9 8 7

+ + + + + 

6) (10 10 10 ) 555 19 18 17 + +  21) ab +ba11

7) (2 2 2 ..... 2 ) 3;7;15 2 3 60 + + + +  22) (abc −cba)99

8) (1 5 5 5 ..... 5 5 5 ) 31 2 3 96 97 98 + + + + + + +  23) (5n + 7)( 4n + 6)2 ∨ n

9) (2 2 2 ..... 2 ) 3

2 3 8

+ + + +  24) (8n +1)(6n +5) kh«ng chia hÕt cho 2 víi mäi n

10) (5 5 5 ..... 5 ) 6

2 3 8

+ + + + 

11) (3 3 3 ... 3 ) 13 2 3 9

+ + + + 

12) (1 7 7 7 .... 7 ) 13 2 3 101 + + + + + 

13) (5 5 5 ) 31 2005 2004 2003 + + 

Chuyªn ®Ò BDHS chøng minh chiia hÕt NguyÔn Thanh Hïng Tr êng THCS Tiªn NHa n¨m

2007