, 10140; -, 100 . ..

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18.03.2013

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, abc, bca cab.

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:

2(100a + 10b + c) = (100b + 10c + a) + (100c + 10a + b),

7a = 3b + 4c, 3(a - b) = 4(c - a). , a - b 4, .. a - b = 4m. 3(a - b) = 4(c - a) , c - a = 3m. a - b = 4m c - a = 3m c - b = 7m. |c - b| ≤ 9, m -1, 0, 1. m = 1, c - b = 7m , : 1) c = 7, b = 0; 2) c = 8, b = 1; 3) c = 9, b = 2. c - a = 3m, a 4, 5, 6. -: 407, 518, 629. , m = -1 370, 481, 592. , m = 0 a - b = 4m = 0, c - a = 3m = 0, .. a = b = c. 9 : 111, 222, ... ,999.

: 407, 518, 629, 370, 481, 592, 111, 222, 333, 444, 555, 666, 777, 888, 999.


abca ( ), (5c + 1)2.

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, 0 < a ≤ 9; 0 ≤ b ≤ 9; 0 ≤ c ≤ 9; a, b, c Z :

1001a + 100b + 10c = 25c2 + 10c + 1;

25c2 = 1001a + 100b - 1;

25c2 - 1000a - 100b = a - 1;

25, a - 1 25. - a = 1.

25c2 - 100b = 1000;

c2 = 40 + 4b;

c = 2.

b + 10 , , b, b = 6. c = 8. 1681.

: 1681.


p - . , 8p2 + 1 - , p = 3.

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p = 3, 8p2 + 1 = 73 - . , p ≠ 3, 8p2 + 1 - .

, 3 . p 3. p = 3k 1. 8p2 + 1 = 8(3k 1)2 + 1 = 8(9k2 6k + 1) + 1 = 72k2 48k + 9 = 3(24k2 16k + 3). .., p = 3k 1, 8p2 + 1 - . p = 3, 8p2 + 1 - .

.


x2 - y2 = 93 .

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(x; y), , .

, :

(x - y)(x + y) = 93;

: 1, 3, 31, 93, -1, -3, -31, -93, . :

, (47; 46), (47; -46), (-47; 46), (-47; -46), (17; 14), (17; -14), (-17; 14), (-17; -14).

: (x; y) (47; 46), (47; -46), (-47; 46), (-47; -46), (17; 14), (17; -14), (-17; 14), (-17; -14).


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