C\u00e2u tr\u1ea3 l\u1eddi \u0111\u01a1n gi\u1ea3n nh\u1ea5t l\u00e0 trong n + 1 s\u1ed1 t\u1ef1 nhi\u00ean l\u1edbn h\u01a1n nhau kh\u00f4ng qu\u00e1 2n l\u1ea7n nh\u1ea5t \u0111\u1ecbnh s\u1ebd c\u00f3 hai s\u1ed1 c\u1ea1nh nhau, hai s\u1ed1 c\u1ea1nh nhau t\u1ea5t nhi\u00ean ph\u1ea3i l\u00e0 c\u00e1c s\u1ed1 nguy\u00ean t\u1ed1 c\u00f9ng nhau. Hai s\u1ed1 c\u1ea1nh nhau n\u1ebfu c\u00f3 \u01b0\u1edbc s\u1ed1 chung l\u00e0 p th\u00ec p nh\u1ea5t \u0111\u1ecbnh ph\u1ea3i b\u1eb1ng 1. Th\u1ebf t\u1ea1i sao trong n + 1 s\u1ed1 t\u1ef1 nhi\u00ean kh\u00f4ng l\u1edbn h\u01a1n nhau qu\u00e1 2n l\u1ea7n nh\u1ea5t \u0111\u1ecbnh ph\u1ea3i c\u00f3 hai s\u1ed1 c\u1ea1nh nhau? Theo \u0111i\u1ec1u ki\u1ec7n \u0111\u1eb7t ra trong t\u1eadp h\u1ee3p t\u1eeb c\u00e1c s\u1ed1 t\u1ef1 nhi\u00ean s\u1ed1 c\u00e1c s\u1ed1 nguy\u00ean t\u1ed1 ph\u1ea3i nh\u1ecf h\u01a1n ho\u1eb7c c\u00f9ng l\u1eafm l\u00e0 b\u1eb1ng 2n. V\u1ea3 l\u1ea1i trong t\u1eadp h\u1ee3p kh\u00f4ng c\u00f3 c\u00e1c s\u1ed1 c\u1ea1nh nhau th\u00ec s\u1ed1 c\u00e1c s\u1ed1 nguy\u00ean t\u1ed1 t\u1ed1i \u0111a ch\u1ec9 l\u00e0 n. V\u00ed d\u1ee5 c\u00e1c t\u1eadp h\u1ee3p kh\u00f4ng c\u00f3 c\u00e1c s\u1ed1 c\u1ea1nh nhau l\u00e0 c\u00e1c t\u1eadp h\u1ee3p: {1, 3, 5, \u20262n – 1} ho\u1eb7c {2, 4, 6, \u20262n}. N\u1ebfu ta l\u1ea1i th\u00eam v\u00e0o c\u00e1c t\u1eadp h\u1ee3p tr\u00ean m\u1ed9t s\u1ed1 n\u00e0o \u0111\u00f3 theo th\u1ee9 t\u1ef1 c\u00e1c s\u1ed1 t\u1ef1 nhi\u00ean th\u00ec t\u1ea5t nhi\u00ean ph\u1ea3i l\u00e0 s\u1ed1 c\u1ea1nh nhau c\u1ee7a n + 1 s\u1ed1 trong m\u1ed7i t\u1eadp h\u1ee3p v\u00e0 t\u1eadp h\u1ee3p m\u1edbi s\u1ebd l\u00e0 t\u1eadp h\u1ee3p c\u00f3 c\u00e1c s\u1ed1 c\u1ea1nh nhau. Ng\u01b0\u1eddi ch\u1ee9ng minh lu\u1eadn \u0111\u1ec1 n\u00e0y l\u00e0 nh\u00e0 to\u00e1n h\u1ecdc Hungari Potard l\u00fac \u00f4ng m\u1edbi 12 tu\u1ed5i.<\/p>\n\n\n\n
B\u00e0i to\u00e1n \u201cH\u00e0n T\u00edn \u0111i\u1ec3m binh\u201d l\u00e0 th\u1ebf n\u00e0o?<\/p>\n\n\n\n
B\u00e0i to\u00e1n \u201cH\u00e1n T\u00edn \u0111i\u1ec3m binh\u201d l\u00e0 m\u1ed9t tr\u00f2 ch\u01a1i d\u1ef1 \u0111o\u00e1n s\u1ed1 th\u00fa v\u1ecb. Gi\u1ea3 s\u1eed b\u1ea1n c\u1ea7m trong tay m\u1ed9t s\u1ed1 l\u00e1 c\u1edd (tr\u00ean d\u01b0\u1edbi 100 l\u00e1), tr\u01b0\u1edbc h\u1ebft b\u1ea1n ch\u1eadp th\u00e0nh nh\u00f3m 3 l\u00e1, s\u1ebd c\u00f2n s\u1ed1 d\u01b0 khi s\u1ed1 c\u00f2n l\u1ea1i kh\u00f4ng \u0111\u1ee7 3 l\u00e1 ; sau \u0111\u00f3 l\u1ea1i ch\u1eadp th\u00e0nh nh\u00f3m 5 l\u00e1 ghi l\u1ea5y s\u1ed1 d\u01b0 \u1edf nh\u00f3m kh\u00f4ng \u0111\u1ee7 5; cu\u1ed1i c\u00f9ng ch\u1eadp th\u00e0nh c\u00e1c nh\u00f3m c\u00f3 7 l\u00e1, ghi l\u1ea5y s\u1ed1 \u1edf nh\u00f3m kh\u00f4ng \u0111\u1ee7 7 l\u00e1. D\u1ef1a v\u00e0o s\u1ed1 l\u00e1 c\u1edd d\u01b0 \u1edf c\u00e1c nh\u00f3m ng\u01b0\u1eddi ta c\u00f3 th\u1ec3 \u0111o\u00e1n s\u1ed1 l\u00e1 c\u1edd \u0111\u00e3 c\u00f3.<\/p>\n\n\n\n
V\u00ed d\u1ee5: Khi ch\u1eadp 3 d\u01b0 1 l\u00e1, ch\u1eadp 5 d\u01b0 2 l\u00e1, ch\u1eadp 7 d\u01b0 1 l\u00e1, v\u1eady c\u00f3 bao nhi\u00eau l\u00e1 c\u1edd?<\/p>\n\n\n\n
Ph\u01b0\u01a1ng ph\u00e1p gi\u1ea3i kh\u00e1 \u0111\u01a1n gi\u1ea3n v\u00e0 ngay t\u1eeb th\u1eddi c\u1ed5 \u0111\u1ea1i \u1edf Trung Qu\u1ed1c \u0111\u00e3 c\u00f3 l\u1eddi gi\u1ea3i. V\u00e0o th\u1eddi nh\u00e0 T\u1ed1ng Chu M\u1eadt g\u1ecdi l\u00e0 \u201cB\u00e0i to\u00e1n Qu\u1ec9 c\u1ed1c\u201d ho\u1eb7c \u201cTo\u00e1n C\u00e1ch t\u01b0\u1eddng\u201d, D\u01b0\u01a1ng Huy g\u1ecdi l\u00e0 \u201cb\u00e0i to\u00e1n ch\u00e9m \u1ed1ng\u201d nh\u01b0ng t\u00ean g\u1ecdi b\u00e0i to\u00e1n \u201cH\u00e0n T\u00edn \u0111i\u1ec3m binh\u201d l\u00e0 t\u00ean g\u1ecdi ph\u1ed5 bi\u1ebfn nh\u1ea5t. C\u00e1ch gi\u1ea3i \u0111\u01b0\u1ee3c tr\u00ecnh b\u00e0y trong quy\u1ec3n s\u00e1ch to\u00e1n c\u1ed5 \u201cT\u00f4n t\u1eed to\u00e1n kinh\u201d. V\u1ec1 sau, T\u1ea7n C\u1eefu Thi\u1ec1u th\u1eddi nh\u00e0 T\u1ed1ng \u0111\u00e3 c\u1ea3i ti\u1ebfn v\u00e0 ph\u1ed5 bi\u1ebfn r\u1ed9ng r\u00e3i v\u1edbi t\u00ean \u201cThu\u1eadt to\u00e1n \u0110\u1ea1i di\u1ec5n\u201d (gi\u1ea3ng gi\u1ea3i v\u1ec1 c\u00e1ch t\u00ednh to\u00e1n). \u0110\u00f3 ch\u00ednh l\u00e0 n\u1ed9i dung m\u00e0 trong l\u1ecbch s\u1eed to\u00e1n h\u1ecdc ng\u01b0\u1eddi ta g\u1ecdi l\u00e0 \u0111\u1ecbnh l\u00ed \u201cth\u1eb7ng d\u01b0 Trung Qu\u1ed1c\u201d, m\u1ed9t b\u00e0i to\u00e1n kh\u00e1 n\u1ed5i ti\u1ebfng.<\/p>\n\n\n\n
N\u1ed9i dung c\u1ee7a ph\u01b0\u01a1ng ph\u00e1p gi\u1ea3i b\u00e0i to\u00e1n c\u00f3 th\u1ec3 bi\u1ec3u di\u1ec5n b\u1eb1ng bi\u1ec3u th\u1ee9c d\u01b0\u1edbi \u0111\u00e2y: a x 70 + b x 21 + c x 15 – 105 trong \u0111\u00f3 a, b, c l\u00e0 c\u00e1c s\u1ed1 d\u01b0 t\u01b0\u01a1ng \u1ee9ng khi ch\u1eadp 3, ch\u1eadp 5 v\u00e0 ch\u1eadp 7 c\u00e1c l\u00e1 c\u1edd. N\u1ebfu con s\u1ed1 t\u00ednh \u0111\u01b0\u1ee3c l\u1edbn h\u01a1n 105 th\u00ec tr\u1eeb cho 105 \u0111\u1ebfn khi \u0111\u01b0\u1ee3c m\u1ed9t s\u1ed1 nh\u1ecf h\u01a1n 105 th\u00ec d\u1eebng l\u1ea1i. Theo c\u00e1ch gi\u1ea3i n\u00e0y b\u00e0i to\u00e1n \u0111o\u00e1n s\u1ed1 l\u00e1 c\u1edd \u1edf tr\u00ean \u0111\u00e2y s\u1ebd c\u00f3 \u0111\u00e1p \u00e1n 1 x 70 + 2 x 21 + 2 x 15 – 105 = 37 l\u00e1.<\/p>\n\n\n\n
Th\u1ebf t\u1ea1i sao trong b\u00e0i to\u00e1n \u201cH\u00e0n T\u00edn \u0111i\u1ec3m binh\u201d l\u00e0 c\u1ea7n b\u1ed9 ba s\u1ed1 3, 5 v\u00e0 7, li\u1ec7u c\u00f3 th\u1ec3 d\u00f9ng c\u00e1c b\u1ed9 ba s\u1ed1 kh\u00e1c \u0111\u01b0\u1ee3c kh\u00f4ng? \u0110\u1ec3 tr\u1ea3 l\u1eddi c\u00e2u h\u1ecfi n\u00e0y ta c\u1ea7n nghi\u00ean c\u1ee9u k\u0129 c\u00e1ch gi\u1ea3i b\u00e0i to\u00e1n \u201cH\u00e0n T\u00edn \u0111i\u1ec3m binh\u201d: \u201c70a + 21b+ 15c – 105\u201d.<\/p>\n\n\n\n
Ta c\u1ea7n xem x\u00e9t c\u00e1c m\u1ed1i quan h\u1ec7 c\u1ee7a 4 s\u1ed1 70, 21, 15 v\u00e0 105 v\u1edbi c\u00e1c s\u1ed1 3, 5, 7. 70 = 2 x 5 x 7; 70 = 3 x 23 + 1 n\u00ean 70 l\u00e0 b\u1ed9i s\u1ed1 chung c\u1ee7a 5 v\u00e0 7 v\u00e0 khi chia cho 3 th\u00ec d\u01b0 1. 21 l\u00e0 b\u1ed9i s\u1ed1 chung c\u1ee7a 3 v\u00e0 7, 21 chia cho 5 th\u00ec d\u01b0 1. 15 l\u00e0 b\u1ed9i s\u1ed1 chung c\u1ee7a 3 v\u00e0 5, 15 chia cho 7 d\u01b0 1. 105 l\u00e0 b\u1ed9i s\u1ed1 chung nh\u1ecf nh\u1ea5t c\u1ee7a ba s\u1ed1 3, 5, 7.<\/p>\n\n\n\n
D\u1ef1a v\u00e0o m\u1ed1i quan h\u1ec7 tr\u00ean \u0111\u00e2y th\u00ec \u201c70a + 21b + 15c – 105\u201d ch\u00ednh l\u00e0 s\u1ed1 ph\u1ea3i t\u00ecm. B\u1edfi v\u00ec: 70a + 21b + 15c – 105 = = (3 x 23 +1) x 1 + (3 x 7 x 2) + (3 x 5 x 2) – (3 x 5 x 7) = 3 x 23 x 1 + 1 x 1+ 3 x 7 x 2 + 3 x 5 x 2 – 3 x 5 x 7 = 3 x (23 x 1 + 7 x 2 + 5 x 2 – 5 x 7) + 1<\/p>\n\n\n\n
V\u00ec v\u1eady 70a + 21b + 15c – 105 chia cho 3 c\u00f3 s\u1ed1 d\u01b0 l\u00e0 1. C\u0169ng l\u00ed lu\u1eadn t\u01b0\u01a1ng t\u1ef1 \u0111em s\u1ed1 n\u00e0y chia cho 5 v\u00e0 cho 7 \u0111\u1ec1u c\u00f3 s\u1ed1 d\u01b0 l\u00e0 2.<\/p>\n\n\n\n
Th\u1ebf t\u1ea1i sao trong b\u00e0i to\u00e1n \u201cH\u00e0n T\u00edn \u0111i\u1ec3m binh\u201d ng\u01b0\u1eddi ta l\u1ea1i d\u00f9ng b\u1ed9 ba s\u1ed1 3, 5, 7. Ch\u00fang ta bi\u1ebft r\u1eb1ng hai s\u1ed1 b\u1ea5t k\u00ec trong ba s\u1ed1 l\u00e0 c\u00e1c s\u1ed1 nguy\u00ean t\u1ed1 t\u1eebng \u0111\u00f4i m\u1ed9t (s\u1ed1 nguy\u00ean t\u1ed1 c\u00f9ng nhau, ch\u1ec9 c\u00f3 \u01b0\u1edbc s\u1ed1 chung l\u00e0 1). T\u1eeb \u0111\u00f3 n\u1ebfu t\u00ecm \u0111\u01b0\u1ee3c m\u1ed9t s\u1ed1 c\u00f3 t\u00ednh ch\u1ea5t l\u00e0 b\u1ed9i s\u1ed1 chung c\u1ee7a hai trong b\u1ed9 ba s\u1ed1 v\u00e0 khi \u0111em chia cho s\u1ed1 th\u1ee9 ba m\u00e0 c\u00f3 s\u1ed1 d\u01b0 l\u00e0 1 nh\u01b0 c\u00e1c s\u1ed1 70, 21, 15 th\u00ec \u0111\u00e1p \u1ee9ng y\u00eau c\u1ea7u c\u1ee7a b\u00e0i to\u00e1n \u201cH\u00e0n T\u00edn \u0111i\u1ec3m binh\u201d.<\/p>\n\n\n\n
Th\u1ebf v\u1edbi c\u00e1c s\u1ed1 kh\u00f4ng nguy\u00ean t\u1ed1 c\u00f9ng nhau th\u00ec c\u00f3 th\u1ec3 t\u00ecm \u0111\u01b0\u1ee3c c\u00e1c s\u1ed1 70, 21 v\u00e0 15 hay kh\u00f4ng? V\u00ed d\u1ee5 ch\u1ecdn ba s\u1ed1 4, 6, 7 trong \u0111\u00f3 hai s\u1ed1 4 v\u00e0 6 kh\u00f4ng nguy\u00ean t\u1ed1 c\u00f9ng nhau, c\u00f3 \u01b0\u1edbc s\u1ed1 chung l\u1edbn nh\u1ea5t l\u00e0 2. M\u00e0 b\u1ed9i s\u1ed1 chung c\u1ee7a c\u00e1c s\u1ed1 6, 7 \u0111\u1ec1u l\u00e0 c\u00e1c s\u1ed1 ch\u1eb5n n\u1ebfu \u0111em chia cho s\u1ed1 4 th\u00ec \u0111\u1ec1u c\u00f3 s\u1ed1 d\u01b0 l\u00e0 s\u1ed1 ch\u1eb5n m\u00e0 kh\u00f4ng th\u1ec3 l\u00e0 s\u1ed1 1, v\u00ec v\u1eady ch\u00fang ta s\u1ebd kh\u00f4ng t\u00ecm \u0111\u01b0\u1ee3c s\u1ef1 t\u01b0\u01a1ng h\u1ee3p v\u1edbi 70, 21, 15. N\u00ean b\u00e0i to\u00e1n \u201cH\u00e0n T\u00edn \u0111i\u1ec3m binh\u201d kh\u00f4ng s\u1eed d\u1ee5ng \u0111\u01b0\u1ee3c ba s\u1ed1 kh\u00f4ng nguy\u00ean t\u1ed1 c\u00f9ng nhau.<\/p>\n\n\n\n
Ch\u00fang ta c\u00f3 th\u1ec3 b\u1ecf b\u1ed9 ba s\u1ed1 kh\u00e1c v\u1edbi 3, 5, 7 m\u00e0 d\u00f9ng b\u1ed9 ba s\u1ed1 nguy\u00ean t\u1ed1 c\u00f9ng nhau kh\u00e1c. V\u00ed d\u1ee5 2, 3, 11 bi\u1ec3u th\u1ee9c c\u1ee7a gi\u1ea3i ph\u00e1p l\u00e0 \u201c33a + 22b + 12c – 66\u201d. Trong \u0111\u00f3 c\u00e1c s\u1ed1 33, 22, 12 v\u00e0 66 tho\u1ea3 m\u00e3n 4 m\u1ed1i quan h\u1ec7 nh\u01b0 \u0111\u00e3 n\u00eau \u1edf tr\u00ean v\u00e0 c\u00e1c b\u1ea1n d\u1ec5 d\u00e0ng t\u00ecm th\u1ea5y s\u1ed1 ph\u1ea3i t\u00ecm l\u00e0 37.<\/p>\n\n\n\n
T\u1eeb kho\u00e1: B\u00e0i to\u00e1n \u201cH\u00e0n T\u00edn \u0111i\u1ec3m binh\u201d, \u0111\u1ecbnh l\u00ed th\u1eb7ng d\u01b0 Trung Qu\u1ed1c, s\u1ed1 nguy\u00ean t\u1ed1 t\u1eebng \u0111\u00f4i m\u1ed9t, b\u1ed9i s\u1ed1 chung v\u00e0 b\u1ed9i s\u1ed1 chung nh\u1ecf<\/p>\n\n\n\n
<\/p>\n\n\n\n\n\n\n","protected":false},"excerpt":{"rendered":"
C\u00e2u tr\u1ea3 l\u1eddi \u0111\u01a1n gi\u1ea3n nh\u1ea5t l\u00e0 trong n + 1 s\u1ed1 t\u1ef1 nhi\u00ean l\u1edbn h\u01a1n nhau kh\u00f4ng qu\u00e1 2n l\u1ea7n nh\u1ea5t \u0111\u1ecbnh s\u1ebd c\u00f3 hai s\u1ed1 c\u1ea1nh nhau, hai s\u1ed1 c\u1ea1nh nhau t\u1ea5t nhi\u00ean ph\u1ea3i l\u00e0 c\u00e1c s\u1ed1 nguy\u00ean t\u1ed1 c\u00f9ng nhau. Hai s\u1ed1 c\u1ea1nh nhau n\u1ebfu c\u00f3 \u01b0\u1edbc s\u1ed1 chung l\u00e0 p th\u00ec […]<\/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-7513","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\/7513","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=7513"}],"version-history":[{"count":0,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/posts\/7513\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/media?parent=7513"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/categories?post=7513"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/tags?post=7513"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}