3. A positive integer is called fully divisible if it is divisible by each of its digits. These digits must also all be different (and not equal to 0). For example, 162 is fully divisible, because it is divisible by 1, 6, and 2.

How many fully divisible two-digit numbers are there?
A) 4
B) 5
C) 6
D) 7
E) 8
A3. B) 5

First, look at the numbers that start with the digit 1. A number is always divisible by 1, so this condition is always met. Now we will check which of the numbers 12, 13, 14, 15, 16, 17, 18, and 19 are also divisible by their last digit, and we see that 12 and 15 are.

Now look at the numbers starting with 2. To be divisible by 2, the last digit of the number must be even. So we only have the possibilities 24, 26, and 28, and we see that only 24 is fully divisible. For the numbers starting with 3, we have the possibilities 36 and 39 (the others are not divisible by 3), and we see that only 36 is fully divisible. For the numbers starting with 4, only 48 is a possibility, and we see that it is also fully divisible. For numbers where the first digit is 5 or greater, there is no second digit such that the number is divisible by the first digit. For example, for the first digit 5, 55 would have been the only possibility, but it is not allowed because the two digits must be different.

In total, there are 5 fully divisible two-digit numbers: $12, 15, 24, 36$, and 48.