题目信息
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 ?
A:142
B:71
C:99
D:20
E:18
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正确答案：
E:18
Arithmetic Place value
Since abc is numerically equal to 100a + 10b + c and cba is numerically equal to 100c + 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.
Since abc is numerically equal to 100a + 10b + c and cba is numerically equal to 100c + 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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