初中奧數 急

設原數為10x + y
改變後數為10a + b
0 <= a,b,x,y,<= 9
10x + y + 2 = 10a + b ... (1)
x + y = 2(a + b) ... (2)
(1) - (2) => 9x + 2 = 8a - b
x = (8a - b - 2) / 9
當a = 0, 找不到適的b
當a = 1, b = 6, x = 0, y = 14 (不合)
當a = 2, b = 5, x = 1, y = 13 (不合)
當a = 3, b = 4, x = 2, y = 12 (不合)
當a = 4, b = 3, x = 3, y = 11 (不合)
當a = 5, b = 2, x = 4, y = 10 (不合)
當a = 6, b = 1, x = 5, y = 9
當a = 7, b = 0, x = 6, y = 8
當a = 7, b = 9, x = 5, y = 27 (不合)
當a = 8, b = 8, x = 6, y = 26 (不合)
當a = 9, b = 7, x = 7, y = 25 (不合)
原數為59, 改變後數為61
原數為68, 改變後數為70
2) 設abcde = x
3 * 1abcde = 3*(100000 + x) = 300000 + 3x
abcde1 = 10x + 1
300000 + 3x = 10x + 1
299999 = 7x
x = 42857
1abcde = 142857




3) 設四位數為X = ABCD = 1000A + 100B + 10C + D ... (1)
(A + C) - (B + D) = 11n其中n為整數 ... (2)
(1) => X = 1001A - A + 11C - C + 99B + B + D
X = 11(91A + C) + (B + D) - (A + C)
X = 11(91A + C) - 11n
X = 11(91A + C - n)為11倍數

4) abc = 37n其中n為整數
def = 37m其中m為整數
abcdef
= abc *1000 + def
= 37n*1000 + 37m
= 37(100n + m) 可被37整除

2009-10-07 01:33:46 補充：
改正(4) abc + def = 37m
abcdef = abc*1000 + def
= abc + def + 999*abc
= 37m + abc * 27 * 37
= 37(m + 27*abc)可被37整除

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