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다음 문제를 풀려니 시간이 오래 걸리네요. 간단히 접근할 수 있는 방법이 있을까요?
9 digits number = ABCD5EFGH가 있습니다.– The first number is divisible by 1.– The first two digits is divisible by 2.– The first three digits is divisible by 3.– And so on.– With no 0’sWhat is the nine digits number?
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See below:
Let’s say the number is ABCDEFGHI (each letter is one of the digits from 1 to 9). The rules say that A has to be divisible by 1, AB by 2, etc. Here are the observations:
1. Whatever the final number, the rules for A and I will always be satisfied (every digit is divisible by 1, and the sum of all the digits is divisible by 9).
2. E must be 5.
3. B, D, F and H must be even digits. Since there are only 4 even digits anyway, they must belong in these positions. Therefore, A, C, G and I must be the remaining (1, 3, 7 and 9).
4. Since ABC is divisible by 3, to get ABCDEF divisible by 6, we need DEF also divisible by 3. I.e., D + E + F must be divisible by 3. Of all the combinations possible for DEF (254, 256, 258, 452, 456, 458, 652, 654, 658, 852, 854 and 856), only four are valid: 258, 852, 456 and 654.
5. CD must be divisible by 4. No matter what C is (whether 1, 3, 7 or 9), C8 and C4 do not work. So, DEF can only be 258 or 654.
6. FGH must be divisible by 8. Again, trying all combinations possible (814, 816, 834, 836, 874, 876, 894, 896, 412, 418, 432, 438, 472, 478, 492 and 498), the only valid ones are: 816, 896, 432 and 472.
7. So, the number can only be one of these four:
A4C25816I, A4C25896I, A8C65432I and A8C65472I.8. A + B + C must be divisible by 3. Trying out all possible combinations for A and I, and eliminating invalid ones, we are left with these 10 numbers:
147258963, 741258963, 189654327, 981654327, 789654321, 987654321, 183654729, 381654729, 189654723 and 981654723.9. Quickly checking the last condition (divisibility by 7) against each of the 10, we see that only 381654729 fits.
(Reference: http://discuss.joelonsoftware.com/default.asp?interview.11.330437.
