Question

Find the smallest positive integer which ends in 17, is divisible by 17, and whose digits sum to 17.

Answer 1

This integer is x17, or xx17, or xxx17
all the digits make a sum of 17, ?+1+7=17, ?=9. the other digits need to make a sum of 9
of the numbers from 1-9, only 1 multiplying 7 will result in a 7 in the one's place, so my reasoning is that the ending 17 needs to remain independent, that is, we need to look for the first digits that will make a sum of 9 AND divisible by 17. Keep counting by 17, 
17, 34, 51, 68, 85, 102, 119, 136, 153
we can see that the first sum of 9 happens when 17*9=153, 1+5+3=9
so the smallest integer that satisfies all the conditions is 15317

Please let me know if you find another way to figure it out, or if there is a smaller interger