Question

I am a number that is greater than 600 but less than 700. The sum of my digits is 18 and I'm not divisible by 2, 5,7, or 11. What number am I?

Answer 1

I thought about this for a while and then recognized that this number is possibly divisible by {3, 4, 6, 8, 9, 10, 12, etc.}

So I look for possible sets of 3 numbers here that add up to 18.  The first such set I recognized was {4, 6, 8}.  These digits do add up to 18.  Unfortunately, 468 is not in the desired range (600,700).  

Thus, I could try either {6, 4, 8} or {6, 8, 4}.

There may be other combinations of 3 numbers (which must add up to 18).

Test 684.  Is 684 between 600 and 700, not including 600 or 700?  YES.

Is the sum of {6, 4, 8} equal to 18?  YES.   So, 648 is one possible answer.  Are there others?