Numbers on a board

Here is a gentle thinker from the 2014 Grey Kangaroo:

Dean’s teacher asks him to write several different positive integers on the board. Exactly two of them are to be divisible by 2 and exactly 13 of them are to be divisible by 13. M is the greatest of these numbers.

What is the least possible value of M?

We must write down 13 multiples of 13, at most two of which may be even. So we must write down at least 11 odd multiples of 13.

The 11 smallest odd multiples of 13 are 13, 39, 65, 91, 117, 143, 169, 195, 221, 247, 273.

It follows that our list must contain a number at least as big as 273. But by picking two more multiples of 13, both even and both less than 273 (such as 26 and 52), we can in fact satisfy all the requirements given with M=273. Thus the least possible value of M is 273.

This is easy to trip over if you rush into things, but very quick with careful thinking.

Short and sweet.

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