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BI TP S HC (Bi s 1)

PHP CHIA HT.

1.nh l. Cho ,a b l cc s nguyn ty v 0b , khi c hai s nguyn ,q r duy nht sao cho : a bq r vi 0 r b .

Ta ni a l s b chia, b l s chia, q l thng v r l s d, s d r c th nhn b gi tr l 0,1,..., 1b .

Khi 0r th a bq , khi ta ni a chia ht cho b hay b l c ca a , k hiu a b hay |b a .

2.Tnh cht.

a) Nu a b v b c th a c .

b) Nu a b v b a th a b hoc a b

c)Nu a c v b c th a b c

3.Cc v d.

a)V d 1. Chng minh rng trong 1n n s nguyn lin tip ch c mt v ch mt s chia ht cho n .

Gii.

Gi s n s nguyn lin tip l 1, 2,...,m m m n

Ly m chia cho n ta c m nq r vi 0 1r n

Khi ta c : m r nq m n r nq n n vi 1 n r n .

Do c s trong n s nguyn lin tip chia ht cho n, ta cn chng minh s l duy nht.

Tht vy gi s c hai s m i v m j cng chia ht cho n , vi 1 ,i j n

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Khi : 0m i m j i j n i j i j (v i j n )

Vy trong n s nguyn lin tip c duy nht mt s chia ht cho n.

b) V d 2. Chng minh rng trong hai s chn lin tip c mt v ch mt s chia ht cho 4.

Gii.

Gi hai s chn lin tip l , 2a a .

+Nu 4 4a a k , khi 2 4 2 2 2 1a k k 4

+Nu a4 m do a chn nn 4 2 2 4 1 4a k a k

c)V d 3. Cho a l mt s nguyn. Chng minh

a) 3 11 6a a b) 1 2 1 6a a a c) 5 30a a d) 42 16 30a a

Gii.

a) 3 211 12 1 1 12 6a a a a a a a a a

v 1, , 1a a a l 3 s nguyn lin tip nn tch 1 1 6a a a

nn 3 11 6a a

b) 1 2 1 1 2 1 1 1 1 2a a a a a a a a a a a a a

c) 5 4 21 1 1 1a a a a a a a a

Ta cn chng minh 21 1 1 5a a a a

Xt 5 ,5 1,5 2,5 3,5 4a k k k k k ta c iu phi chng minh.

d) 4 52 16 30 2a a a a a

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4. Bi tp.

Bi 1. Chng minh rng trong hai s chn lin tip c mt v ch mt s chia ht cho 4.

Bi 2. Cho a l mt s nguyn. Chng minh

a) 3 11 6a a b) 1 2 1 6a a a c) 5 30a a d) 42 16 30a a

Bi 3. Cho m l mt s nguyn ty . Chng minh

a) 4 3 23 14 21 10 24m m m m b) 5 4 3 25 5 5 6 120m m m m m

Bi 4. Cho m l mt s nguyn l. Chng minh

a) 3 23 3 48m m m b) 12 8 4 1 512m m m

Bi 5. Chng minh rng tng lp phng ca 3 s nguyn lin tip chia ht cho 9.

Bi 6. Cho ,m n l hai s chnh phng l lin tip. Chng minh 1 192mn m n

Bi 7. Cho , ,a b n l cc s nguyn, 0,n a b . Chng minh

a) n na b a b b) n na b a b vi n l c) n na b a b vi n chn.

(gi : dng phng php chng minh qui np)

Bi 8. Chng minh

a) 1011 1 10 b) 5555 22222222 5555 7

Bi 9. a)Vi n l s t nhin chn, chng minh 20 16 3 1 133n n n

b)Vi 1n l s t nhin l, chng minh 46 296.13 1947n n

Bi 10. Chng minh 2903 803 464 261 1897n n n n n

Bi 11. Chng minh rng vi n l s t nhin

a) 2 2 111 12 133n n b) 2 2 15 26.5 8 59n n n c) 27.5 12.6 19n n

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Bi 12. Cho a l s nguyn khng chia ht cho 5 v khng chia ht cho 7. Chng minh 4 4 21 15 1 35a a a Bi 13. Chng minh rng ch s tn cng ca cc s t nhin n v 5n l nh nhau.

Bi 14. a) Cho 2n v n nguyn t cng nhau vi 6. Chng minh 2 1 24n

b)Cho n l v nguyn t cng nhau vi 3. Chng minh 4 1 48n

c)Cho n l v nguyn t cng nhau vi 5. Chng minh 4 1 80n

Bi 15. a)Chng t rng s 1983 19170,3 1983 1917A l mt s nguyn.

b) 1994 19920.8 1994 1992B c l mt s nguyn hay khng ?

Bi 16. Chng minh rng vi s t nhin 1n v k l mt s t nhin l ta c

1 2 3 ... 1 2 3 ...k k k kn n

Bi 17. Cho 2 1 1 2 1 12 2 1; 2 2 1n n n nn na b

Chng minh rng vi mi s t nhin n c mt v ch mt s trong hai s ,n na b chia ht cho 5.

Bi 18. Chng minh rng vi 1n , ta c

a) 21 1nn n b) 22 12 1 2 1nn n

Bi 19. Chng minh rng vi 1n v k l mt s t nhin l, ta c

2 21 2n nk

Bi 20. a)Tm tt c s t nhin n sao cho 2 1n chia ht cho 7.

b)Chng minh rng vi mi s t nhin n, 2 1n khng chia ht cho 7.