A two digit number is four times the sum and three times the product of its digits. Find the number:

Question

Accepted Answer

To solve the problem, we need to find a two-digit number that satisfies two conditions: it is four times the sum of its digits and three times the product of its digits. Let's analyze each option to determine why Option 4 (24) is the correct answer.

### Explanation for Option 4 (24):

1. **Sum of the digits**: The digits of the number 24 are 2 and 4. The sum of these digits is 2 + 4 = 6.
2. **Four times the sum**: Four times the sum of the digits is 4 × 6 = 24. This matches the number itself.
3. **Product of the digits**: The product of the digits is 2 × 4 = 8.
4. **Three times the product**: Three times the product of the digits is 3 × 8 = 24. This also matches the number itself.

Since both conditions are satisfied, Option 4 (24) is indeed the correct answer.

### Explanation for why other options are incorrect:

1. **Option 1 (12)**:
   - **Sum of the digits**: 1 + 2 = 3
   - **Four times the sum**: 4 × 3 = 12 (This matches the number, but let's check the second condition)
   - **Product of the digits**: 1 × 2 = 2
   - **Three times the product**: 3 × 2 = 6 (This does not match the number)

2. **Option 2 (22)**:
   - **Sum of the digits**: 2 + 2 = 4
   - **Four times the sum**: 4 × 4 = 16 (This does not match the number)
   - **Product of the digits**: 2 × 2 = 4
   - **Three times the product**: 3 × 4 = 12 (This does not match the number)

3. **Option 3 (14)**:
   - **Sum of the digits**: 1 + 4 = 5
   - **Four times the sum**: 4 × 5 = 20 (This does not match the number)
   - **Product of the digits**: 1 × 4 = 4
   - **Three times the product**: 3 × 4 = 12 (This does not match the number)

In conclusion, Option 4 (24) is the only number that satisfies both conditions: it is four times the sum of its digits and three times the product of its digits. Therefore, Option 4 is the correct answer.