Vi\u1ec7c ph\u00e1n \u0111o\u00e1n v\u1ec1 t\u00ednh chia h\u1ebft c\u1ee7a m\u1ed9t s\u1ed1 t\u1ef1 nhi\u00ean cho m\u1ed9t s\u1ed1 t\u1ef1 nhi\u00ean kh\u00e1c l\u00e0 m\u1ed9t y\u00eau c\u1ea7u th\u01b0\u1eddng g\u1eb7p trong cu\u1ed9c s\u1ed1ng. \u0110\u01b0\u01a1ng nhi\u00ean n\u1ebfu trong tay b\u1ea1n c\u00f3 m\u1ed9t m\u00e1y t\u00ednh, b\u1ea1n ch\u1ec9 c\u1ea7n \u0111\u1eb7t m\u1ed9t ph\u00e9p t\u00ednh h\u1ee3p l\u00fd l\u00e0 t\u00ednh to\u00e1n xong. Khi s\u1ed1 chia l\u00e0 s\u1ed1 \u0111\u01a1n gi\u1ea3n (v\u00ed d\u1ee5 s\u1ed1 c\u00f3 m\u1ed9t ch\u1eef s\u1ed1) th\u00ec c\u00f3 th\u1ec3 d\u00f9ng m\u1ed9t s\u1ed1 quy t\u1eafc ph\u00e1n \u0111o\u00e1n. Khi c\u00e1c b\u1ea1n n\u1eafm \u0111\u01b0\u1ee3c c\u00e1c quy t\u1eafc th\u00ec kh\u00f4ng c\u1ea7n c\u00f3 m\u00e1y t\u00ednh, b\u1ea1n c\u0169ng c\u00f3 th\u1ec3 gi\u1ea3i b\u00e0i to\u00e1n v\u1ec1 t\u00ednh chia h\u1ebft kh\u00e1 nhanh ch\u00f3ng.<\/p>\n\n\n\n
Quy t\u1eafc ph\u00e1n \u0111o\u00e1n v\u1ec1 t\u00ednh chia h\u1ebft c\u00f3 hai lo\u1ea1i: M\u1ed9t l\u00e0, xem ch\u1eef s\u1ed1 cu\u1ed1i ho\u1eb7c m\u1ea5y ch\u1eef s\u1ed1 cu\u1ed1i c\u1ee7a c\u00e1c con s\u1ed1 nh\u01b0 \u1edf c\u00e1c m\u1ee5c 1 v\u00e0 2, sau \u0111\u00e2y; hai l\u00e0 t\u00ednh t\u1ed5ng c\u00e1c ch\u1eef s\u1ed1 trong con s\u1ed1 ho\u1eb7c xem x\u00e9t c\u00e1c h\u1ec7 s\u1ed1 th\u00edch h\u1ee3p cho c\u00e1c t\u1ed5ng m\u00e0 ph\u00e1n \u0111o\u00e1n nh\u01b0 \u1edf c\u00e1c m\u1ee5c t\u1eeb 3 \u0111\u1ebfn 6.<\/p>\n\n\n\n
M\u1ed9t s\u1ed1 t\u1ef1 nhi\u00ean l\u00e0 s\u1ed1 l\u1ebb s\u1ebd kh\u00f4ng chia h\u1ebft cho 2; m\u1ed9t s\u1ed1 ch\u1eb5n chia h\u1ebft cho 2. V\u00ed d\u1ee5 c\u00e1c s\u1ed1 0, 2, 4. 6,\u2026s\u1ebd chia h\u1ebft cho 2, c\u00f2n c\u00e1c s\u1ed1 l\u1ebb nh\u01b0 1,3, 5, 7,\u2026kh\u00f4ng chia h\u1ebft cho 2.<\/p>\n\n\n\n
M\u1ed9t s\u1ed1 t\u1ef1 nhi\u00ean s\u1ebd chia h\u1ebft cho 5 n\u1ebfu ch\u1eef s\u1ed1 cu\u1ed1i c\u1ee7a s\u1ed1 \u0111\u00f3 l\u00e0 s\u1ed1 0 ho\u1eb7c 5; m\u1ed9t s\u1ed1 t\u1ef1 nhi\u00ean chia h\u1ebft cho 25 n\u1ebfu hai ch\u1eef s\u1ed1 cu\u1ed1i c\u1ee7a s\u1ed1 \u0111\u00f3 l\u00e0 00, 25, 50 ho\u1eb7c 75, v\u00ed d\u1ee5 s\u1ed1 120795 c\u00f3 th\u1ec3 chia h\u1ebft cho 5 nh\u01b0ng kh\u00f4ng chia h\u1ebft cho 25.<\/p>\n\n\n\n
M\u1ed9t s\u1ed1 chia h\u1ebft cho 3 khi t\u1ed5ng c\u00e1c ch\u1eef s\u1ed1 c\u1ee7a s\u1ed1 \u0111\u00f3 chia h\u1ebft cho<\/p>\n\n\n\n
M\u1ed9t s\u1ed1 chia h\u1ebft cho 9 n\u1ebfu t\u1ed5ng c\u00e1c ch\u1eef s\u1ed1 c\u1ee7a s\u1ed1 \u0111\u00f3 chia h\u1ebft cho 9. V\u00ed nh\u01b0 s\u1ed1 147345 th\u00ec t\u1ed5ng c\u00e1c ch\u1eef s\u1ed1 c\u1ee7a s\u1ed1 \u0111\u00f3 l\u00e0 5 + 4 + 3 + 7 + 4+ 1 = 24 chia h\u1ebft cho 3 m\u00e0 kh\u00f4ng chia h\u1ebft cho 9 n\u00ean s\u1ed1 tr\u00ean ch\u1ec9 chia h\u1ebft cho 3 m\u00e0 kh\u00f4ng chia h\u1ebft cho 9.<\/p>\n\n\n\n
V\u00ec sao l\u1ea1i c\u00f3 quy t\u1eafc d\u1ef1 \u0111o\u00e1n kh\u00e1 \u0111\u01a1n gi\u1ea3n nh\u01b0 v\u1eady? Gi\u1ea3 s\u1eed cho s\u1ed1:<\/p>\n\n\n\n
A = a0 + 10a1 + 102a2 + 103a3 + \u2026 trong \u0111\u00f3 a0, a1, a2, a3\u2026l\u00e0 ch\u1eef s\u1ed1 h\u00e0ng \u0111\u01a1n v\u1ecb, h\u00e0ng ch\u1ee5c, h\u00e0ng tr\u0103m, h\u00e0ng ngh\u00ecn\u2026c\u1ee7a s\u1ed1 A; ta c\u00f3 th\u1ec3 vi\u1ebft:<\/p>\n\n\n\n
A = a0 + 10a1 + 102a2 + 103a3 + \u2026 = [ (10 – 1) a1 + (102 – 1)a2 + (103 -1) a3] + (a0 + a1 + a2 + a3 +\u2026).<\/p>\n\n\n\n
D\u1ec5 d\u00e0ng nh\u1eadn th\u1ea5y 10n-1 l\u00e0 b\u1ed9i s\u1ed1 c\u1ee7a 3 v\u00e0 9 v\u00ec v\u1eady n\u1ebfu s\u1ed1 h\u1ea1ng th\u1ee9 hai c\u1ee7a bi\u1ec3u th\u1ee9c s\u1ed1 A (bi\u1ec3u th\u1ee9c trong ngo\u1eb7c \u0111\u01a1n) vi\u1ebft \u1edf tr\u00ean l\u00e0 b\u1ed9i s\u1ed1 c\u1ee7a 3 v\u00e0 9 th\u00ec s\u1ed1 A s\u1ebd chia h\u1ebft cho 3 v\u00e0 9. T\u1eeb \u0111\u00f3 ta \u0111i \u0111\u1ebfn quy t\u1eafc n\u1ebfu a0 + a1 + a2 + a3 + \u2026 l\u00e0 b\u1ed9i s\u1ed1 c\u1ee7a 3 ho\u1eb7c 9 th\u00ec s\u1ed1 A chia h\u1ebft cho 3 ho\u1eb7c 9.<\/p>\n\n\n\n
M\u1ed9t s\u1ed1 chia h\u1ebft cho 4 n\u1ebfu t\u1ed5ng c\u1ee7a ch\u1eef s\u1ed1 h\u00e0ng \u0111\u01a1n v\u1ecb v\u00e0 ch\u1eef s\u1ed1 h\u00e0ng ch\u1ee5c nh\u00e2n \u0111\u00f4i chia h\u1ebft cho 4 th\u00ec s\u1ed1 \u0111\u00f3 chia h\u1ebft cho 4. M\u1ed9t s\u1ed1 t\u1ef1 nhi\u00ean chia h\u1ebft cho 8 n\u1ebfu t\u1ed5ng c\u1ee7a ch\u1eef s\u1ed1 h\u00e0ng \u0111\u01a1n v\u1ecb c\u1ed9ng v\u1edbi ch\u1eef s\u1ed1 h\u00e0ng ch\u1ee5c nh\u00e2n \u0111\u00f4i v\u00e0 ch\u1eef s\u1ed1 h\u00e0ng tr\u0103m nh\u00e2n 4 chia h\u1ebft cho 8 th\u00ec s\u1ed1 \u0111\u00f3 chia h\u1ebft cho 8. V\u00ed d\u1ee5 s\u1ed1 1390276 chia h\u1ebft cho 4 v\u00ec 6 + 2 x 7 = 20 chia h\u1ebft cho 4 n\u00ean s\u1ed1 1390276 chia h\u1ebft cho 4. S\u1ed1 1390276 kh\u00f4ng chia h\u1ebft cho 8 v\u00ec theo quy t\u1eafc 6 + 2 x 7 + 4 x 2 = 28 kh\u00f4ng chia h\u1ebft cho 8.<\/p>\n\n\n\n
C\u00e1ch ch\u1ee9ng minh quy t\u1eafc v\u1eeba n\u00eau c\u0169ng t\u01b0\u01a1ng t\u1ef1 nh\u01b0 c\u00e1ch ch\u1ee9ng minh \u1edf 3.<\/p>\n\n\n\n
Ta vi\u1ebft v\u00ed d\u1ee5:<\/p>\n\n\n\n
A = [ (10 – 2) a1 + (102 – 4)a2 + 103a3 + \u2026] +(a0 + 2a1 + 4a2).<\/p>\n\n\n\n
D\u1ec5 d\u00e0ng nh\u1eadn th\u1ea5y bi\u1ec3u th\u1ee9c trong ngo\u1eb7c vu\u00f4ng l\u00e0 b\u1ed9i s\u1ed1 c\u1ee7a 8 v\u00e0 A s\u1ebd chia h\u1ebft cho 8 n\u1ebfu h\u1ea1ng s\u1ed1 th\u1ee9 hai c\u1ee7a A ph\u00eda b\u00ean ph\u1ea3i (bi\u1ec3u th\u1ee9c trong ngo\u1eb7c \u0111\u01a1n) l\u00e0 b\u1ed9i s\u1ed1 c\u1ee7a 8.<\/p>\n\n\n\n
M\u1ed9t s\u1ed1 chia h\u1ebft cho 11 n\u1ebfu hi\u1ec7u s\u1ed1 c\u1ee7a t\u1ed5ng c\u00e1c s\u1ed1 ch\u1eb5n v\u00e0 t\u1ed5ng c\u00e1c ch\u1eef s\u1ed1 h\u00e0ng l\u1ebb l\u00e0 b\u1ed9i s\u1ed1 c\u1ee7a 11. V\u00ed d\u1ee5 v\u1edbi s\u1ed1 268829 t\u1ed5ng c\u00e1c ch\u1eef s\u1ed1 \u1edf h\u00e0ng l\u1ebb 9 + 8 + 6 = 23, t\u1ed5ng c\u00e1c ch\u1eef s\u1ed1 h\u00e0ng ch\u1eb5n l\u00e0 2 + 8 + 2 = 12 hi\u1ec7u c\u1ee7a ch\u00fang \u0111\u00fang b\u1eb1ng 11 n\u00ean s\u1ed1 n\u00e0y s\u1ebd chia h\u1ebft cho 11. L\u1ea1i nh\u01b0 v\u1edbi s\u1ed1 1257643 th\u00ec hi\u1ec7u c\u1ee7a hai t\u1ed5ng c\u00e1c ch\u1eef s\u1ed1 l\u00e0 (3 + 6 + 5 + 1) – (4 + 7 + 2) = 2. V\u00ec kh\u00f4ng ph\u1ea3i l\u00e0 b\u1ed9i s\u1ed1 c\u1ee7a 11 n\u00ean s\u1ed1 n\u00e0y kh\u00f4ng chia h\u1ebft cho 11. \u0110\u1ec3 ch\u1ee9ng minh quy t\u1eafc ta vi\u1ebft:<\/p>\n\n\n\n
A = [ (10 + 1)a1 + (102 – 1)a2 + (103 + 1)a3 + (104 – 1)a4 +\u2026] + [(a0 + a2 +\u2026) – (a1 + a3 + \u2026)].<\/p>\n\n\n\n
S\u1ed1 h\u1ea1ng th\u1ee9 nh\u1ea5t c\u1ee7a A l\u00e0 b\u1ed9i s\u1ed1 c\u1ee7a 11 n\u00ean n\u1ebfu s\u1ed1 h\u1ea1ng th\u1ee9 hai l\u00e0 b\u1ed9i s\u1ed1 c\u1ee7a 11 (hi\u1ec7u c\u1ee7a t\u1ed5ng c\u00e1c ch\u1eef s\u1ed1 \u1edf h\u00e0ng ch\u1eb5n v\u00e0 c\u00e1c ch\u1eef s\u1ed1 \u1edf h\u00e0ng l\u1ebb) \u0111\u01b0\u01a1ng nhi\u00ean l\u00e0 A s\u1ebd chia h\u1ebft cho 11.<\/p>\n\n\n\n
Ch\u1ee9ng minh quy t\u1eafc chia h\u1ebft cho 7 kh\u00e1 ph\u1ee9c t\u1ea1p m\u00e0 \u00fd ngh\u0129a th\u1ef1c ti\u1ec5n l\u1ea1i h\u1ea1n ch\u1ebf n\u00ean \u1edf \u0111\u00e2y ch\u1ec9 gi\u1edbi thi\u1ec7u quy t\u1eafc m\u00e0 kh\u00f4ng \u0111i s\u00e2u v\u00e0o c\u00e1ch ch\u1ee9ng minh.<\/p>\n\n\n\n
B\u1ea1n h\u00e3y nh\u1edb k\u0129 d\u00e3y h\u1ec7 s\u1ed1 tu\u1ea7n ho\u00e0n sau \u0111\u00e2y: 1, 2, 3, -1, -2, -3, 1, 3, 2,\u2026<\/p>\n\n\n\n
Mu\u1ed1n ph\u00e1n \u0111o\u00e1n v\u1ec1 t\u00ednh chia h\u1ebft c\u1ee7a m\u1ed9t s\u1ed1 t\u1ef1 nhi\u00ean b\u1ea5t k\u00ec c\u00f3 chia h\u1ebft cho 7 hay kh\u00f4ng c\u00e1c b\u1ea1n h\u00e3y nh\u00e2n c\u00e1c ch\u1eef s\u1ed1 v\u1edbi d\u00e3y s\u1ed1 \u0111\u00e3 n\u00eau, sau \u0111\u00f3 t\u00ednh t\u1ed5ng s\u1ed1 c\u1ee7a ch\u00fang. V\u00ed d\u1ee5, b\u1ea1n h\u00e3y nh\u00e2n c\u00e1c ch\u1eef s\u1ed1 b\u1eaft \u0111\u1ea7u t\u1eeb ch\u1eef s\u1ed1 \u0111\u01a1n v\u1ecb l\u00e0 h\u1ec7 s\u1ed1 1, ch\u1eef s\u1ed1 h\u00e0ng ch\u1ee5c l\u00e0 h\u1ec7 s\u1ed1 3, ch\u1eef s\u1ed1 h\u00e0ng tr\u0103m v\u1edbi h\u1ec7 s\u1ed1 2, ch\u1eef s\u1ed1 h\u00e0ng ng\u00e0n v\u1edbi h\u1ec7 s\u1ed1 -1, v.v. r\u1ed3i t\u00ednh t\u1ed5ng \u0111\u1ea1i s\u1ed1 c\u1ee7a c\u00e1c t\u00edch thu \u0111\u01b0\u1ee3c. N\u1ebfu t\u1ed5ng s\u1ed1 v\u1eeba t\u00ednh \u0111\u01b0\u1ee3c chia h\u1ebft cho 7 th\u00ec s\u1ed1 \u0111\u00f3 s\u1ebd chia h\u1ebft cho 7. V\u00ed d\u1ee5 x\u00e9t s\u1ed1 5125764 chia h\u1ebft cho 7 v\u00ec: 4 + 2 x 6 + 2 x 7 – 5- 3 x 2 -2 x 1 + 5 = 28 chia h\u1ebft cho 7.<\/p>\n\n\n\n
Khi x\u00e9t t\u00ednh chia h\u1ebft c\u1ee7a m\u1ed9t s\u1ed1 t\u1ef1 nhi\u00ean ta c\u1ea7n ch\u00fa \u00fd \u0111\u1ebfn t\u00ednh ch\u1ea5t quan tr\u1ecdng sau \u0111\u00e2y: N\u1ebfu m\u1ed9t s\u1ed1 A \u0111\u1ed3ng th\u1eddi chia h\u1ebft cho hai s\u1ed1 p v\u00e0 q th\u00ec c\u0169ng chia h\u1ebft cho t\u00edch s\u1ed1 p x q c\u1ee7a hai s\u1ed1. V\u00ed d\u1ee5 s\u1ed1 5125764 \u0111\u1ed3ng th\u1eddi chia h\u1ebft cho hai s\u1ed1 7 v\u00e0 4 n\u00ean s\u1ed1 n\u00e0y s\u1ebd chia h\u1ebft cho t\u00edch s\u1ed1 7 x 4 = 28 v.v\u2026<\/p>\n\n\n\n
T\u1eeb kho\u00e1: T\u00ednh chia h\u1ebft.<\/p>\n\n\n\n
<\/p>\n\n\n\n\n\n\n","protected":false},"excerpt":{"rendered":"
Vi\u1ec7c ph\u00e1n \u0111o\u00e1n v\u1ec1 t\u00ednh chia h\u1ebft c\u1ee7a m\u1ed9t s\u1ed1 t\u1ef1 nhi\u00ean cho m\u1ed9t s\u1ed1 t\u1ef1 nhi\u00ean kh\u00e1c l\u00e0 m\u1ed9t y\u00eau c\u1ea7u th\u01b0\u1eddng g\u1eb7p trong cu\u1ed9c s\u1ed1ng. \u0110\u01b0\u01a1ng nhi\u00ean n\u1ebfu trong tay b\u1ea1n c\u00f3 m\u1ed9t m\u00e1y t\u00ednh, b\u1ea1n ch\u1ec9 c\u1ea7n \u0111\u1eb7t m\u1ed9t ph\u00e9p t\u00ednh h\u1ee3p l\u00fd l\u00e0 t\u00ednh to\u00e1n xong. Khi s\u1ed1 chia l\u00e0 s\u1ed1 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_disable_autopaging":false},"categories":[15,9],"tags":[],"class_list":["post-7487","post","type-post","status-publish","format-standard","hentry","category-hoi-dap-khoa-hoc","category-toan-hoc"],"_links":{"self":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/posts\/7487","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/comments?post=7487"}],"version-history":[{"count":0,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/posts\/7487\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/media?parent=7487"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/categories?post=7487"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/tags?post=7487"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}