L thuyt ng d v mt s ng dngA. T VN I- L DO CHN TI Ton hc l mt b mn khoa hc rt tru tng, c suy lun mt cch lgic v l nn tng cho vic nghin cu cc b mn khoa hc khc. S hc l mt phn khng th thiu v n chim mt vai tr kh quan trng trong b mn ny. L thuyt chia ht trong vnh s nguyn l mt ni dung kh quan trng trong phn s hc. Hn na, y cng l mng rt kh khn cho gio vin v hc sinh trong qu trnh dy v hc. Xut pht t vn , ti tm ti, nghin cu, trao i v hc hi bn b, ng ch ng nghip v tm ra cha kho gii quyt vn ny. l l thuyt ng d. Nm hc 20092010, ti c s phn cng ca cc ng ch trong t v lm chuyn cm vn ny c nhiu ng nghip quan tm v chia s. V vy ti chn L thuyt ng d v mt s ng dung lm sng kin kinh nghim nhm trao i vi bn b ng nghip nhiu hn v lnh vc ny. II- C S KHOA HC V PHNG PHP NGHIN CU 1. C s l lun. L thuyt ng d c xy dng trn nn tng l php chia trn vnh s nguyn. L mt ni dung c suy lun mt cch lgic, cht ch. Trn c s l thuyt ng d c hai nh bc l le v Fcma a ra 2 nh l rt ni ting v c tnh ng dng rt cao. 2. C s thc tin L thuyt ng d s cho ta phng php ng d, l mt ng tc c tnh cht k thut gip chng ta b sung gii quyt vn chia ht trong vnh s nguyn. 3. Phng php nghin cu - Nghin cu ti liu - Trao i qua ng nghip

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L thuyt ng d v mt s ng dng

B. NI DUNGI- NG D THC 1. nh ngha v cc iu kin: a. nh ngha: Cho m N *; a,b Z. Nu a v b khi chia cho m c cng s d ta ni: a v b ng d theo mun m. K hiu: a b (mod m) H thc: a b (mod m) gi l ng d thc. V d: 19 3 (mod 8); 123-25 3 (mod 4) b. Cc iu kin tng ng: a b (mod m) (a - b) mt Z sao cho: a = b + m.t.

2. Cc tnh cht a. Quan h ng d l mt quan h tng ng trn tp hp Z c ngha l: 123a a (mod m) a b (mod m) => b a (mod m) a b (mod m); b c (mod m) => a c (mod m)

b. Ta c th cng tng v mt vi nhau theo cng mt mun. C th: ai bi (mod m) i = => (mod m)

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L thuyt ng d v mt s ng dng

c. Ta c th nhn tng v vi nhau nhiu ng d thc theo cng mt mun. C th: ai bi (mod m);i = => (mod m);

3. Cc h qu a. a b (mod m) => a c b c (mod m) b. a + c b (mod m) => a b - c (mod m) c. a b (mod m) => a + k.m b (mod m) d. a b (mod m) => a.c b.c (mod m)

e. a b (mod m) => an bn (mod m) n N f. Cho f(x) = an xn + an-1 xn-1 + . . . +a1x + a0 ai Z . Nu (mod m) th

ta cng c f( ) f( ) (mod m) 1. Ta ) 0 (mod m) th ta cng c: c bit: f( cth nhn c hai v v mun ca mt + k.m) thc vi m) f( ng d 0 (modk Z cng mt s nguyn dng.

g.

C th l: Ta c th chia c hai(mod m) => ng d(mod m.c)mt c chung ca a b v ca mt a.c b.c thc cho chngTa c tht vic hai v v mun ca mt ng d thc vi 2. nguyn chia mun. C th mt c dng ca chng. cng l:

h.

a.c b.c C th l: (mod m); CLN (c; m) =1 => a b (mod m) a b (mod m); 0 < c C (a; b; m) => a/c b/c (mod m/c) 4

L thuyt ng d v mt s ng dng

k. Nu 2 s a v b ng d vi nhau tho nhiu mun th chng ng d vi nhau theo mun l bi chung nh nht ca mun y. C th l: a b (mod mi), i = 1, n => a b (mod m). Trong : m = BCNN(m1, m2 mn) l. Nu a v b ng d vi nhau theo mun m th chng cng ng d vi nhau theo mun l c dng ca m. C th l: a b (mod m); 0 < (m) => a b (mod ) u. Nu: a b (mod m) th: CLN( a; m) = CLN( b; m).

II- NH L LE V NH L FCMA

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L thuyt ng d v mt s ng dng1. nh l le a. Hm s le- (m) Cho hm s (m) c xc nh nh sau: - m = 1 ta c: (m) = 1 - m > 1 th (m)l cc s t nhin khng vt qu m 1 v nguyn t vi m b. Cng thc tnh (m) b.1 Ta c: (m) = (p) = p (1 p ) b.2 2 m = p1 p2 p3 ... pn (pi l cc s nguyn t, 1 l s t nhin khc 0 ).1 3 n

m = p ( p l s nguyn t, l s t nhin khc 0)1

Ta c: c.

(m) = m (1 p )(1 p )(1 p )(1 p ) 1 2 3 n

1

1

1

1

nh l le

Cho m l mt s t nhin khc 0 v a l mt s nguyn t vi m. Khi y ta c: (mod m)

2.

nh l Fcma

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L thuyt ng d v mt s ng dng

- nh l Fcma 1 Cho p l mt s t nhin khc v a l mt s nguyn khng chia ht cho m. Khi y ta c:

ap - 1

1 (mod p)

- nh l Fcma 2 Cho p l mt s nguyn t, a l mt s nguyn bt k. Khi y ta c:

ap - 1

a (mod p)

III - MT S NG DNG 1. Tm s d trong php chia V d1: Tm s d trong php chia: 29455 3 chia cho 9 Gii: Ta c: 2945 2 (mod 9) => 29455 3 25 3 (mod 9) M 25 3 2 (mod 9) Vy s d ca 29455 3 chia cho 9 l 2

V d 2: Tm s d trong php chia 109345 chia cho 14

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L thuyt ng d v mt s ng dngGii: Ta c: 109 -3 (mod 14) => 109345 (-3)345 (mod 14) Ta li c: ( -3; 14 ) = 1 Hn na: (14) = 14.(1 )(1 ) = 6 Nn: (-3)6 1 (mod 14) (theo nh l le) => (-3)345 (-3)3 (mod 14) Mt khc: (-3)3 = -27 1 (mod 14) Vy s d trong php chia 109345 chia cho 14 l 1 V d 2:Tm s d trong php chia: (19971998 + 19981999 +19992000 )10 chia cho 111 Gii: Ta c: 1998 0 (mod 111) => 1997 -1 (mod 111) v 1999 1 (mod 111) Nn ta c: 19971998 + 19981999 +19992000 2 (mod 111) (19971998 + 19981999 +19992000 )10 210 (mod 111) Mt khc ta c: 210 = 1024 25 (mod 111) Vy (19971998 + 19981999 +19992000 )10 chia cho 111 c s d l 25 2. Chng minh chia ht V d 1: Chng minh: 3100 3 chia ht cho 13 Gii. Ta c: 33 = 27 1 (mod 13) => 3100 = 3.399 3.1 (mod 13) => 3100- 3 0 (mod 13). Vy 3100-3 chia ht cho 131 2 1 7

V d 2: Chng minh 62n + 1 + 5n + 2 chia ht cho 31 vo mi n l s t nhin

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L thuyt ng d v mt s ng dngGii: Ta c: 62 5 (mod 31) => 62n 5n (mod 31) Mt khc: 6 - 52 (mod 31) Nn: 62n + 1 -5n + 2 (mod 31) Vy 62n + 1 + 5n + 2 chia ht cho 31. V d 3: Chng minh 2 34 n +1

+ 3 vi n l s t nhin 111 2 1 5

Gii: Ta c: (11) = 10; (10) = 10(1 )(1 ) = 4. p dng L le ta c: (3; 10) = 1 => 3(10) 1 (mod 10) 34 1 (mod 10) => 34n + 1 3 (mod 10) t 34n + 1 = 10.k + 3 vi k N. Khi ta c: 234 n +1

+ 3 = 210.k +1 + 3

p dng nh l le ta c: (2; 11) = 1 Nn 2(11) 1 (mod 11) 210 1 (mod 11) => 210.k +3 23 (mod 11)4 n +1 => 210.k +3 + 3 23 +3 (mod 11) 23 + 3 0 (mod 11)

Vy 2 3

4 n +1

+ 3 11

3. Tm ch s tn cng V d 1: Tm 2 ch s tn cng ca 20092010 Gii: Ta c: 20092010 92010 (mod 100) p dng nh l le ta c: (9; 100) =1 Nn: 9(100) 1 (mod 100). M (100) = 100.(1 )(1 ) = 40 2 5 Hay: 940 1 (mod 100) => 92010 910 (mod 100) M 910 = 3486784401 1 (mod 100). Vy 2 ch s tn cng ca 20092010 l 01.1 1

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L thuyt ng d v mt s ng dngV d 2: Tm 3 ch s tn cng ca 21954 Gii: Ta thy (2; 1000) = 2 nn cha th p dng trc tip nh l le c. Ta c: (21954; 1000) = 8. Ta xt 21951 chia cho 125 p dng nh l le ta c: (2; 125) = 1 Nn: 2(125) 1 (mod 125). M (125) = 125(1 ) = 25 5 Hay: 225 1 (mod 125) => 21951 2 (mod 125) => 21951. 23 2.23 (mod 125.23) 21954 16 (mod 1000) Vy 3 ch s tn cng ca 21954 l 016 V d 3: Tm 2 ch s tn cng ca 9 9 Gii: p dng nh l le ta c: (9; 100) = 1; (100) = 40; => 940 1 (mod 100). (*) Mt khc ta c: 92 1 (mod 40) => 99 9 (mod 40). t 99 = 40.k + 9 vi k N(9

1

**)

9 T (*) v (**) suy ra: 9 9 99 (mod 100)

M: 99 = 387420489 89 (mod 100) Vy 2 ch s tn cng ca 9 9 l 899

C. KT LUN V KIN NGH

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L thuyt ng d v mt s ng dngI- KT LUN L thuyt ng d l mt mng kin thc kh rng v tng i phc tp. Tuy nhin kh nng ng dng ca n th rt rng v c tnh u vit cao. N phc v rt nhiu trong qu trnh ging dy mn Ton THCS. Hn na t l thuyt ng d m s cho ta cc lnh vc khc. V d nh: Phng trnh v nh, L thuyt chia ht trong vnh a thc Z(x), ... V vy trong sng kin kinh nghim ny ti khng th a ra ht c. L thuyt ng d v mt s ng dng l mt iu ti ang nung nu v hon thin hn na. Trong sng kin ny chc chn cn nhiu vn cha y . V vy ti knh mong qu v, bn b ng nghip gp chia s cng hon thin hn. II- KIN KIN NGH Trong nhiu nm qua, cng vi s quan tm gip ca cc c quan v nhiu mt cho ngnh gio dc v cng vi s pht trin nhanh ca cng ngh thong tin. Cc sng kin kinh nghim ca nhiu gio vin cng ngy cng c cht lng. Tuy nhin kh nng trao i, phm vi ng dng cha c rng ri v nhiu tng hay cha n vi tt c mi gio vin v hc sinh nhm bin cc tng thnh hin thc. V vy ti knh mong Phng GD-T, T chuyn mn phng nn to iu kin cc sng kin, tng hay c th n vi tt c cc gio vin v hc sinh, bng cch c th in n, a ln trang Web ni b ca phng cc tng tr thnh hin thc v c ngha hn.

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