Problem:
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with terms 247,475 , and 756 and end with the term 824. Let be the sum of all the terms in the sequence. What is the largest prime number that always divides
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Solution:
A given digit appears as the hundreds digit, the tens digit, and the units digit of a term the same number of times. Let be the sum of the units digits in all the terms. Then , so must be divisible by . To see that need not be divisible by any larger prime, note that the sequence 123, 231, 312 gives .
The problems on this page are the property of the MAA's American Mathematics Competitions