If 142N is divisible by 12, then what is the value of N?

Options

A. 2
B. 4
C. 6
D. 8

Answer

To find the value of \( N \) such that \( 142N \) is divisible by 12, we can follow these steps:

### Step 1: Understand the divisibility rule
A number is divisible by 12 if it is divisible by both 3 and 4.

### Step 2: Check divisibility by 4
For a number to be divisible by 4, the last two digits of the number must form a number that is divisible by 4. In our case, the last two digits are \( 42N \).

We can check the values of \( N \) from the options provided (2, 4, 6, 8):

- If \( N = 2 \): \( 42N = 42 \) (not divisible by 4)
- If \( N = 4 \): \( 42N = 44 \) (divisible by 4)
- If \( N = 6 \): \( 42N = 46 \) (not divisible by 4)
- If \( N = 8 \): \( 42N = 48 \) (divisible by 4)

So, the possible values of \( N \) that make \( 142N \) divisible by 4 are \( N = 4 \) and \( N = 8 \).

### Step 3: Check divisibility by 3
For a number to be divisible by 3, the sum of its digits must be divisible by 3. We will check the sum of the digits for \( 142N \).

- For \( N = 4 \):
- The number is \( 1424 \)
- Sum of digits: \( 1 + 4 + 2 + 4 = 11 \) (not divisible by 3)

- For \( N = 8 \):
- The number is \( 1428 \)
- Sum of digits: \( 1 + 4 + 2 + 8 = 15 \) (divisible by 3)

### Step 4: Conclusion
Since \( 1428 \) is divisible by both 4 and 3, the value of \( N \) that makes \( 142N \) divisible by 12 is \( N = 8 \).

### Final Answer
The value of \( N \) is \( 8 \).

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