Video Explanation

Text Explanation

This is a very tricky question.  First, let's just try a few numbers to see if what kind of patterns emerge.

Now, try some with an even tens-digit:

OK, so right away, we can eliminate (A), because K is sometimes odd. Also, the M = 12 case produces K = 33, so we can eliminate (E).  Certainly, after this initial search, it would appear that (D) is the correct answer, but we can't be sure just yet. 

Notice that if the sum of the digits of M is S, then it appears that K = 11 × S.  That pattern fits every number we have tried above.  For a square, let's try a value for M for which the sum of digits is 11.   

That's a perfect square!  Therefore, we can eliminate (B).  Now, let's try a value for M for which the sum of digits is 13. 

That's a number divisible by 13.  Therefore, we can eliminate (C).  Now we can be certain that the answer, as we expected, is (D).