Ch\u00fang ta \u0111\u00e3 bi\u1ebft c\u00e1ch t\u00ednh s\u1ed1 tr\u1eadn \u0111\u1ea5u theo th\u1ec3 th\u1ee9c thi \u0111\u1ea5u v\u00f2ng tr\u00f2n. Th\u1ebf nh\u01b0ng vi\u1ec7c s\u1eafp x\u1ebfp l\u1ecbch thi \u0111\u1ea5u th\u1ebf n\u00e0o \u0111\u1ec3 c\u00e1c \u0111\u1ea5u th\u1ee7 c\u00f3 th\u1ec3 g\u1eb7p c\u00e1c \u0111\u1ea5u th\u1ee7 kh\u00e1c nhau trong c\u00e1c v\u00f2ng \u0111\u1ea5u? Ta xem x\u00e9t v\u00ed d\u1ee5 v\u1ec1 c\u00e1c \u0111\u1ed9i n\u1eef trong cu\u1ed9c thi \u0111\u1ea5u b\u00f3ng b\u00e0n trong \u0111\u00f3 c\u00f3 hai b\u1ea3ng: m\u1ed9t b\u1ea3ng c\u00f3 s\u00e1u \u0111\u1ed9i, m\u1ed9t b\u1ea3ng b\u1ea3y \u0111\u1ed9i. Ta th\u1eed s\u1eafp x\u1ebfp l\u1ecbch thi \u0111\u1ea5u cho b\u1ea3ng c\u00f3 s\u00e1u \u0111\u1ed9i, s\u00e1u \u0111\u1ed9i n\u00e0y thi \u0111\u1ea5u theo th\u1ec3 th\u1ee9c \u0111\u1ea5u v\u00f2ng tr\u00f2n m\u1ed9t l\u01b0\u1ee3t. K\u00ed hi\u1ec7u x l\u00e0 s\u1ed1 phi\u00ean hi\u1ec7u c\u00e1c \u0111\u1ed9i x \uf0ce{1, 2, \u2026,6}, r k\u00ed hi\u1ec7u x v\u00f2ng thi \u0111\u1ea5u r \uf0ce{1, 2, \u2026,5} nh\u01b0 v\u1eady m\u1ed7i \u0111\u1ed9i ph\u1ea3i ti\u1ebfn h\u00e0nh n\u0103m v\u00f2ng \u0111\u1ea5u. D\u01b0\u1edbi \u0111\u00e2y l\u00e0 b\u1ea3ng s\u1eafp x\u1ebfp l\u1ecbch thi \u0111\u1ea5u cho s\u00e1u \u0111\u1ed9i trong n\u0103m v\u00f2ng thi \u0111\u1ea5u. Trong b\u1ea3ng c\u00f3 r h\u00e0ng, x c\u1ed9t, s\u1ed1 phi\u00ean hi\u1ec7u m\u1ed7i \u0111\u1ed9i l\u00e0 y, s\u1ed1 v\u00f2ng \u0111\u1ea5u l\u00e0 r.<\/p>\n\n\n\n
B\u1ea3ng l\u1ecbch thi \u0111\u1ea5u \u0111\u01b0\u1ee3c s\u1eafp x\u1ebfp nh\u01b0 th\u1ebf n\u00e0o?<\/p>\n\n\n\n
Tr\u01b0\u1edbc h\u1ebft xin gi\u1edbi thi\u1ec7u kh\u00e1i ni\u1ec7m \u201c\u0111\u1ed3ng d\u01b0\u201d. V\u1edbi hai s\u1ed1 nguy\u00ean a, b, n\u1ebfu ch\u1ecdn \u0111\u01b0\u1ee3c m\u1ed9t s\u1ed1 m sao cho khi a, b chia cho m (s\u1ed1 chia) th\u00ec ta \u0111\u01b0\u1ee3c m\u1ed9t th\u01b0\u01a1ng s\u1ed1 l\u00e0 s\u1ed1 nguy\u00ean nh\u01b0ng ph\u00e9p chia c\u00f3 s\u1ed1 d\u01b0 b\u1eb1ng nhau. V\u00ed d\u1ee5 v\u1edbi hai s\u1ed1 a = 34 v\u00e0 b = 12 v\u00e0 n\u1ebfu ch\u1ecdn m = 11 th\u00ec s\u1ed1 d\u01b0 c\u1ee7a hai ph\u00e9p chia b\u1eb1ng nhau v\u00e0 b\u1eb1ng 1. Ng\u01b0\u1eddi ta n\u00f3i a v\u00e0 b c\u00f3 m\u1ed1i li\u00ean quan v\u1edbi nhau qua \u0111\u1ed3ng d\u01b0 m v\u00e0 vi\u1ebft: a \u2261 b (mod m). Ta \u0111\u1ecdc a v\u00e0 b \u0111\u1ed3ng d\u01b0 theo mo\u0111un m. Kh\u00e1i ni\u1ec7m \u0111\u1ed3ng d\u01b0 ra \u0111\u1eddi r\u1ea5t s\u1edbm t\u1eeb th\u1ebf k\u1ec9 th\u1ee9 V. \u1ede Trung Qu\u1ed1c kh\u00e1i ni\u1ec7m \u0111\u1ed3ng d\u01b0 xu\u1ea5t hi\u1ec7n \u0111\u1ea7u ti\u00ean trong b\u1ed9 s\u00e1ch \u201cS\u00e1ch to\u00e1n T\u00f4n T\u1eed\u201d. Trong \u0111\u1eddi s\u1ed1ng h\u1eb1ng ng\u00e0y ch\u00fang ta c\u0169ng th\u01b0\u1eddng g\u1eb7p hi\u1ec7n t\u01b0\u1ee3ng \u0111\u1ed3ng d\u01b0. V\u00ed d\u1ee5 trong m\u1ed9t th\u00e1ng n\u00e0o \u0111\u00f3 n\u1ebfu ng\u00e0y 2 l\u00e0 th\u1ee9 t\u01b0 th\u00ec c\u00e1c ng\u00e0y 9, 16, 23 c\u0169ng l\u00e0 ng\u00e0y th\u1ee9 t\u01b0. V\u00ec th\u1ebf c\u00e1c s\u1ed1 9, 16, 23 li\u00ean quan v\u1edbi nhau qua \u0111\u1ed3ng d\u01b0 theo mo\u0111un 7.<\/p>\n\n\n\n
N\u00f3i chung \u0111\u1ec3 x\u1ebfp l\u1ecbch thi \u0111\u1ea5u theo th\u1ec3 th\u1ee9c thi \u0111\u1ea5u v\u00f2ng tr\u00f2n c\u00f3 N \u0111\u1ed9i tham gia ch\u1ec9 c\u1ea7n \u1edf v\u00f2ng \u0111\u1ea5u th\u1ee9 r ta ch\u1ecdn gi\u00e1 tr\u1ecb y th\u1ebf n\u00e0o cho x + y = r (mod N – 1) l\u00e0 \u0111\u01b0\u1ee3c.<\/p>\n\n\n\n
Nh\u01b0 trong v\u00ed d\u1ee5 tr\u00ean, ta ph\u1ea3i ch\u1ecdn y th\u1ebf n\u00e0o \u0111\u1ec3 x + y chia 5 c\u00f3 s\u1ed1 d\u01b0 b\u1eb1ng r l\u00e0 \u0111\u01b0\u1ee3c.<\/p>\n\n\n\n
V\u00ed d\u1ee5 \u1edf v\u00f2ng \u0111\u1ea5u th\u1ee9 nh\u1ea5t (r = 1, x+y = 6) n\u00ean v\u1edbi c\u00e1c gi\u00e1 tr\u1ecb x = 1; y = 5; x =2; y = 4 th\u00ec \u0111\u1ec1u \u0111\u00e1p \u1ee9ng \u0111\u01b0\u1ee3c y\u00eau c\u1ea7u. Nh\u01b0ng x = 3; y = 3 th\u00ec g\u1eb7p tr\u01b0\u1eddng h\u1ee3p \u0111\u1ed9i th\u1ee9 ba l\u1ea1i \u0111\u1ea5u v\u1edbi ch\u00ednh m\u00ecnh n\u00ean kh\u00f4ng th\u1ec3 \u0111\u01b0\u1ee3c. V\u00ec v\u1eady trong tr\u01b0\u1eddng h\u1ee3p n\u00e0y, ta quy \u01b0\u1edbc ch\u1ecdn \u0111\u1ed9i cu\u1ed1i c\u00f9ng l\u00e0 \u0111\u1ed9i s\u1ed1 6 thi \u0111\u1ea5u v\u1edbi \u0111\u1ed9i 3. Nh\u01b0 v\u1eady \u1edf h\u00e0ng th\u1ee9 nh\u1ea5t ta gi\u1ea3i quy\u1ebft xong.<\/p>\n\n\n\n
\u1ede v\u00f2ng thi \u0111\u1ea5u th\u1ee9 hai (r = 2, x + y = 7), \u1edf h\u00e0ng th\u1ee9 hai kh\u00f4ng g\u1eb7p tr\u1edf ng\u1ea1i g\u00ec.<\/p>\n\n\n\n
\u1ede v\u00f2ng \u0111\u1ea5u th\u1ee9 ba (r = 3; x + y = 8), khi x = 1, y = 7 v\u00ec kh\u00f4ng c\u00f3 \u0111\u1ed9i b\u00f3ng c\u00f3 phi\u00ean hi\u1ec7u n\u00e0y, n\u00ean trong tr\u01b0\u1eddng h\u1ee3p n\u00e0y ta ch\u1ecdn x + y = r th\u00ec x = 1, y = 2; x = 2, y = 1. Sau \u0111\u00f3 l\u1ea1i quay v\u1ec1 x + y = 8 th\u00ec x = 3, y = 5; khi x = 4 th\u00ec y = 4 n\u00ean b\u00e2y gi\u1edd y kh\u00f4ng th\u1ec3 b\u1eb1ng 4 m\u00e0 l\u1ea5y b\u1eb1ng 6.<\/p>\n\n\n\n
B\u1eb1ng c\u00e1ch t\u01b0\u01a1ng t\u1ef1 ng\u01b0\u1eddi ta c\u00f3 th\u1ec3 l\u1eadp l\u1ecbch thi \u0111\u1ea5u cho th\u1ec3 th\u1ee9c thi \u0111\u1ea5u v\u00f2ng tr\u00f2n c\u1ee7a m\u1ed9t b\u1ea3ng c\u00f3 6 \u0111\u1ed9i.<\/p>\n\n\n\n
Nh\u01b0 v\u1eady n\u1ebfu s\u1ed1 c\u00e1c \u0111\u1ed9i ghi t\u00ean thi \u0111\u1ea5u l\u00e0 s\u1ed1 ch\u1eb5n, th\u00ec m\u1ed7i \u0111\u1ed9i trong m\u1ed9t v\u00f2ng \u0111\u1ea5u \u0111\u1ec1u c\u00f3 \u0111\u1ea5u th\u1ee7 kh\u00e1c nhau. Tuy nhi\u00ean \u0111\u00e2y kh\u00f4ng ph\u1ea3i l\u00e0 l\u1ecbch \u0111\u1ea5u duy nh\u1ea5t. N\u1ebfu s\u1ed1 \u0111\u1ed9i tham gia thi \u0111\u1ea5u l\u00e0 s\u1ed1 l\u1ebb, th\u00ec c\u00e1ch x\u1ebfp l\u1ecbch thi \u0111\u1ea5u nh\u01b0 v\u1eeba tr\u00ecnh b\u00e0y s\u1ebd kh\u00f4ng th\u00edch h\u1ee3p.<\/p>\n\n\n\n
T\u1eeb kho\u00e1: Kh\u00e1i ni\u1ec7m \u0111\u1ed3ng d\u01b0.<\/p>\n\n\n\n
<\/p>\n\n\n\n\n\n\n","protected":false},"excerpt":{"rendered":"
Ch\u00fang ta \u0111\u00e3 bi\u1ebft c\u00e1ch t\u00ednh s\u1ed1 tr\u1eadn \u0111\u1ea5u theo th\u1ec3 th\u1ee9c thi \u0111\u1ea5u v\u00f2ng tr\u00f2n. Th\u1ebf nh\u01b0ng vi\u1ec7c s\u1eafp x\u1ebfp l\u1ecbch thi \u0111\u1ea5u th\u1ebf n\u00e0o \u0111\u1ec3 c\u00e1c \u0111\u1ea5u th\u1ee7 c\u00f3 th\u1ec3 g\u1eb7p c\u00e1c \u0111\u1ea5u th\u1ee7 kh\u00e1c nhau trong c\u00e1c v\u00f2ng \u0111\u1ea5u? Ta xem x\u00e9t v\u00ed d\u1ee5 v\u1ec1 c\u00e1c \u0111\u1ed9i n\u1eef trong cu\u1ed9c thi \u0111\u1ea5u […]<\/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-7597","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\/7597","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=7597"}],"version-history":[{"count":0,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/posts\/7597\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/media?parent=7597"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/categories?post=7597"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/tags?post=7597"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}