To solve this problem, we need to find the number of 3-digit numbers formed using the digits 2, 3, 4, 5, and 7, with no repetition of digits allowed, and these numbers should not be divisible by 3. Let's break down the solution step-by-step:

1. Calculating the total number of 3-digit numbers without repetition:

The number of ways to form a 3-digit number from 5 unique digits (2, 3, 4, 5, 7) without repetition can be calculated using permutations:

The total number of permutations for choosing 3 digits out of 5 and arranging them is given by:

$$5P3 = \frac{5!}{(5-3)!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} = 5 \times 4 \times 3 = 60$$

So, there are 60 possible 3-digit numbers that can be formed from the digits 2, 3, 4, 5, and 7 without repetition.

2. Finding the 3-digit numbers divisible by 3:

A number is divisible by 3 if the sum of its digits is divisible by 3. Let's consider the sums of every combination of these three digits to find out which sums are divisible by 3:

Possible sums of combinations:

• 2 + 3 + 4 = 9 (divisible by 3)
• 2 + 4 + 7 = 13
• 2 + 5 + 7 = 14
• 2 + 3 + 5 = 10
• 2 + 3 + 7 = 12 (divisible by 3)
• 3 + 4 + 5 = 12 (divisible by 3)
• 3 + 4 + 7 = 14
• 3 + 5 + 7 = 15 (divisible by 3)
• 4 + 5 + 7 = 16

The combinations whose sums are divisible by 3 are:

• 2, 3, 4
• 2, 3, 7
• 3, 4, 5
• 3, 5, 7

Since the sum of the digits is divisible by 3 for these combinations, any permutation of these sets will yield a number divisible by 3:

The number of 3-digit numbers that can be formed from each set of 3 digits is:

$$3! = 6$$

So, the total number of 3-digit numbers divisible by 3 is:

$$4 \text{ sets} \times 6 \text{ permutations per set} = 24$$

3. Calculating the 3-digit numbers not divisible by 3:

To find the 3-digit numbers not divisible by 3, we subtract the number of those divisible by 3 from the total number of 3-digit numbers:

$$60 - 24 = 36$$

Therefore, the number of 3-digit numbers that can be formed using the digits 2, 3, 4, 5, and 7 without repetition, and which are not divisible by 3, is equal to 36.