How many 3-digit numbers can be formed from the digits 2, 3, 5, 6 and 7, which are divisible by 4 and none of the digits is repeated?

To find the number of 3-digit numbers that can be formed from the digits 2, 3, 5, 6, and 7, which are divisible by 4 and have no repeated digits, we can follow these steps:

Step 1: Determine the possible choices for the hundreds digit. Since none of the digits can be repeated, we have 5 choices for the hundreds digit: 2, 3, 5, 6, or 7.

Step 2: Determine the possible choices for the tens digit. Since none of the digits can be repeated, we have 4 choices for the tens digit. However, we need to consider the divisibility by 4. The only two-digit numbers formed by the given digits that are divisible by 4 are 52, 56, and 72. So, we have 3 choices for the tens digit.

Step 3: Determine the possible choices for the units digit. Since none of the digits can be repeated, we have 3 choices for the units digit. However, we need to consider the divisibility by 4. The only three-digit numbers formed by the given digits that are divisible by 4 are 352, 356, 572, and 576. So, we have 4 choices for the units digit.

Step 4: Calculate the total number of 3-digit numbers. To find the total number of 3-digit numbers, we multiply the number of choices for each digit: 5 choices for the hundreds digit, 3 choices for the tens digit, and 4 choices for the units digit. Therefore, the total number of 3-digit numbers that can be formed is 5 x 3 x 4 = 60.

So, there are 60 3-digit numbers that can be formed from the digits 2, 3, 5, 6, and 7, which are divisible by 4 and have no repeated digits.