Number Theory Puzzle: Competition Problem I have been struggling the past few hours with a problem I initially thought to be easy and simple. Nothing comes to mind besides guessing numbers and the answer. Any help or hints would be greatly appreciated!
Thank You!
Problem: 
If any digit of a given 4-digit number is deleted, the resulting 3-digit number is a divisor of the original number. How many 4-digit numbers have this property? 

We have $14$ such numbers. With the numbers @RossMillikan sir have noted, we also can have $a|ab$ and $b|ab$ case. Which will give numbers like $1200,1500,2400,3600,4800$. The full list is given below: $$ 1100, 1200, 1500, 2200, 2400, 3300, 3600, 4400, 4800, 5500, 6600, 7700, 8800, 9900$$
