Out of the following numbers which is divisible by 132?

Options

A. 31218
B. 78520
C. 38148
D. 52020

Answer

To determine which of the given numbers is divisible by 132, we can follow these steps:

### Step 1: Factor 132
First, we can factor 132 into its prime factors:
\[ 132 = 11 \times 12 = 11 \times (4 \times 3) \]
Thus, we need to check if a number is divisible by 11, 4, and 3.

### Step 2: Check the divisibility rules
1. **Divisibility by 11**: The difference between the sum of the digits at odd places and the sum of the digits at even places should be either 0 or a multiple of 11.
2. **Divisibility by 4**: The last two digits of the number should be divisible by 4.
3. **Divisibility by 3**: The sum of all digits of the number should be divisible by 3.

### Step 3: Analyze each option
Let's analyze each option one by one.

#### Option 1: 31218
- **Last two digits**: 18
- **Check divisibility by 4**: 18 is not divisible by 4 (18 ÷ 4 = 4.5).
- **Conclusion**: Since it is not divisible by 4, it cannot be divisible by 132.

#### Option 2: 78520
- **Last two digits**: 20
- **Check divisibility by 4**: 20 is divisible by 4 (20 ÷ 4 = 5).
- **Sum of digits**: 7 + 8 + 5 + 2 + 0 = 22.
- **Check divisibility by 3**: 22 is not divisible by 3 (22 ÷ 3 = 7.33).
- **Conclusion**: Since it is not divisible by 3, it cannot be divisible by 132.

#### Option 3: 38148
- **Last two digits**: 48
- **Check divisibility by 4**: 48 is divisible by 4 (48 ÷ 4 = 12).
- **Sum of digits**: 3 + 8 + 1 + 4 + 8 = 24.
- **Check divisibility by 3**: 24 is divisible by 3 (24 ÷ 3 = 8).
- **Check divisibility by 11**:
- Odd places: 3 (1st) + 1 (3rd) + 8 (5th) = 12
- Even places: 8 (2nd) + 4 (4th) = 12
- Difference: 12 - 12 = 0 (which is divisible by 11).
- **Conclusion**: Since it is divisible by 4, 3, and 11, it is divisible by 132.

#### Option 4: (Not needed to check)
Since we have already found a number that is divisible by 132, we do not need to check this option.

### Final Answer
The number that is divisible by 132 is **38148**.

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