Answer:
6
Explanation:
You want to know the number of uphill integers divisible by 15, where an uphill integer is one that has its digits strictly increasing.
Divisible by 15
An integer will be divisible by 15 if and only if it is divisible by 3 and 5. An integer is divisible by 5 if it ends in 5 or 0. An uphill integer cannot end in 0, so must end in 5.
An integer is divisible by 3 if the sum of its digits is divisible by 3
Uphill integers
An uphill integer will have differences between successive digits that are positive integers. In order for the final digit to be 5, the sum of these differences must be 5. Hence we can find uphill integers by considering the "integer partitions" of 5: the sets of positive integers whose total is 5. Those sets would be ...
{5}, {4, 1}, {3, 1, 1}, {2, 2, 1}, {2, 1, 1, 1}, {1, 1, 1, 1, 1}
Uphill integers will also have digit differences that are some permutation of each of these sets. For example, the digit differences may be {4, 1} or {1, 4}, corresponding to the numbers 45 or 15.
The uphill integers that end in 5 are ...
5, 45, 15, 35, 25, 345, 145, 125, 245, 235, 135, 2345, 1345, 1245, 1235, 12345
Divisible by 3
We already know each of these is divisible by 5. The ones that have a digit total that is a multiple of 3 are ...
15, 45, 135, 345, 1245, 12345
There are 6 uphill integers divisible by 15.