Let the original n be 10x+y and the new number m be 100x+y. x must be a positive integer and y must be an integer 0-9.
Then (100x+y)/(10x+y) = 10 - (9y)/(10x+y) is an integer.
If y=0 then the fraction is 0 for all x. So the set of all n includes all multiples of 10.
If y=1,2,3,4,6,7,9 then there are no such positive integer x to make (9y)/(10x+y) an integer.
If y=5 then x=1 yields 45/15 = 3. n=15, m=105 is a solution with 105/15 = 10-3.
If y=8 then x=1 yields 72/18 = 4. n=18, m=108 is a solution with 108/18 = 10-4.
So the set of all n consists of 15, 18, and all multiples of 10.