How many 4 digit numbers consisting of only the digits 2,4,6 and 8 are divisible by 4

In order for a number to be divisible by 4, its last two digits must be divisible by 4. This will be the case if either of these conditions holds:

the ones digit is an even multiple of 2, and the tens digit is even
the ones digit is an odd multiple of 2, and the tens digit is odd.

We must count the ways these conditions can be met with the given digits.

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Since we only have even numbers to work with, the ones digit must be an even multiple of 2: 4 or 8. (The tens digit cannot be odd.) The digits 4 and 8 comprise half of the available digits, so half of all possible numbers made from these digits will be divisible by 4.

digits used once

If the numbers must use each digit exactly once, there will be 4! = 24 of them. 24/2 = 12 of these 4-digit numbers will be divisible by 4.

repeated digits

Each of the four digits can have any of four values, so there will be 4^4 = 256 possible 4-digit numbers. Of these, 256/2 = 128 will be divisible by 4.