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  3. The difference 942—249 is a positive multiple of 7． If a，b，and c are nonzero di

The difference 942—249 is a positive multiple of 7． If a，b，and c are nonzero di

游客2025-02-27  23
问题 The difference 942—249 is a positive multiple of 7． If  a，b，and c are nonzero digits，how many 3-digit numbers abc are possible such that the difference abc—cba is a positive multiple of 7?
选项 A、142
B、71
C、99
D、20
E、18
答案 E
解析 Since abc is numerically equal to 100a + 10b + c and cba is numerically equal to 100a + 10b + a, it follows that abc - cba is numerically equal to (100 - 1)a + (10 - 10)b + (1 - 100)c = 99(a - c). Because 7 and 99 are relatively prime, 99(a - c) will be divisible by 7 if and only if a - c is divisible by 7. This leads to two choices for the nonzero digits a and c，namely a = 9, c = 2 and a =8，c=1. For each of these two choices for a and c, b can be any one of the nine nonzero digits. Therefore, there is a total of 2(9) = 18 possible 3-digit numbers abc.
The correct answer is E.
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