perplexus.info :: Just Math : Lasts a year

Lasts a year (Posted on 2011-10-24)

Every 8-digit number structured ABCXABCX, where A,B and C are not necessarily distinct, is divisible by 365 iff X equals either 5 or 0.

Why?

No Solution Yet Submitted by Ady TZIDON

Rating: 3.6667 (3 votes)

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Solution

| Comment 1 of 5

If ABCDABCD is divisible by 365, then it is divisible by 5, so it ends in either 5 or 0. Therefore, X=5 or 0.

Suppose X is either 5 or 0. Then, ABCX is divisible by 5. Also, ABCXABCX=10001ABCX. Let ABCX=5N. Then, ABCXABCX=10001(5N)=50005N=365(137N). Therefore, ABCXABCX is divisible by 365.

Posted by Math Man on 2011-10-24 10:21:21

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