Answer:
- digits used once: 12
- repeated digits: 128
Step-by-step explanation:
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.
<h3>digits used once</h3>
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.
<h3>repeated digits</h3>
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.
