The number of 3-digit numbers, formed using the digits 2, 3, 4, 5 and 7, when the repetition of digits is not allowed, and which are not divisible by 3, is equal to ______________.

Answer

To solve the problem of finding the number of 3-digit numbers formed using the digits 2, 3, 4, 5, and 7, without repetition and which are not divisible by 3, we can follow these steps:

### Step 1: Count the total number of 3-digit numbers without repetition
We have 5 digits: 2, 3, 4, 5, and 7.

- For the hundreds place, we can choose any of the 5 digits.
- For the tens place, we can choose from the remaining 4 digits.
- For the units place, we can choose from the remaining 3 digits.

Thus, the total number of 3-digit numbers is calculated as:
\[ \text{Total} = 5 \times 4 \times 3 = 60 \]

### Step 2: Determine the numbers that are divisible by 3
To find out how many of these numbers are divisible by 3, we need to check the sum of the digits. A number is divisible by 3 if the sum of its digits is divisible by 3.

The digits we have are: 2, 3, 4, 5, and 7.

Calculating the sum of all digits:
\[ 2 + 3 + 4 + 5 + 7 = 21 \]
Since 21 is divisible by 3, we need to find combinations of 3 digits that also yield a sum divisible by 3.

### Step 3: Identify combinations of digits
We can check combinations of three digits from the set {2, 3, 4, 5, 7}:

1. **Combination (2, 3, 4)**:
\[ 2 + 3 + 4 = 9 \quad (\text{divisible by 3}) \]

2. **Combination (2, 3, 5)**:
\[ 2 + 3 + 5 = 10 \quad (\text{not divisible by 3}) \]

3. **Combination (2, 3, 7)**:
\[ 2 + 3 + 7 = 12 \quad (\text{divisible by 3}) \]

4. **Combination (2, 4, 5)**:
\[ 2 + 4 + 5 = 11 \quad (\text{not divisible by 3}) \]

5. **Combination (2, 4, 7)**:
\[ 2 + 4 + 7 = 13 \quad (\text{not divisible by 3}) \]

6. **Combination (2, 5, 7)**:
\[ 2 + 5 + 7 = 14 \quad (\text{not divisible by 3}) \]

7. **Combination (3, 4, 5)**:
\[ 3 + 4 + 5 = 12 \quad (\text{divisible by 3}) \]

8. **Combination (3, 4, 7)**:
\[ 3 + 4 + 7 = 14 \quad (\text{not divisible by 3}) \]

9. **Combination (3, 5, 7)**:
\[ 3 + 5 + 7 = 15 \quad (\text{divisible by 3}) \]

10. **Combination (4, 5, 7)**:
\[ 4 + 5 + 7 = 16 \quad (\text{not divisible by 3}) \]

The combinations that yield a sum divisible by 3 are:
- (2, 3, 4)
- (2, 3, 7)
- (3, 4, 5)
- (3, 5, 7)

This gives us a total of 4 combinations.

### Step 4: Calculate the number of 3-digit numbers for each valid combination
For each valid combination of digits, we can arrange them in:
\[ 3! = 6 \text{ ways} \]
Thus, the total number of 3-digit numbers that are divisible by 3 is:
\[ 4 \times 6 = 24 \]

### Step 5: Calculate the number of 3-digit numbers not divisible by 3
To find the number of 3-digit numbers that are not divisible by 3, we subtract the number of divisible numbers from the total:
\[ \text{Not divisible by 3} = 60 - 24 = 36 \]

### Final Answer
The number of 3-digit numbers formed using the digits 2, 3, 4, 5, and 7, without repetition and which are not divisible by 3, is **36**.

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