For what value of M, 34M is divisible by 12?

Options

A. 2
B. 0
C. 8
D. 6

Answer

To find the value of \( M \) such that \( 34M \) is divisible by 12, we need to ensure that \( 34M \) is divisible by both 3 and 4, since \( 12 = 3 \times 4 \).

### Step 1: Check divisibility by 3
A number is divisible by 3 if the sum of its digits is divisible by 3.

- The digits of \( 34M \) are \( 3, 4, \) and \( M \).
- The sum of the digits is \( 3 + 4 + M = 7 + M \).

Now we need \( 7 + M \) to be divisible by 3.

### Step 2: Find possible values of \( M \)
We can check the possible values of \( M \) (which can be any digit from 0 to 9):

- If \( M = 0 \): \( 7 + 0 = 7 \) (not divisible by 3)
- If \( M = 1 \): \( 7 + 1 = 8 \) (not divisible by 3)
- If \( M = 2 \): \( 7 + 2 = 9 \) (divisible by 3)
- If \( M = 3 \): \( 7 + 3 = 10 \) (not divisible by 3)
- If \( M = 4 \): \( 7 + 4 = 11 \) (not divisible by 3)
- If \( M = 5 \): \( 7 + 5 = 12 \) (divisible by 3)
- If \( M = 6 \): \( 7 + 6 = 13 \) (not divisible by 3)
- If \( M = 7 \): \( 7 + 7 = 14 \) (not divisible by 3)
- If \( M = 8 \): \( 7 + 8 = 15 \) (divisible by 3)
- If \( M = 9 \): \( 7 + 9 = 16 \) (not divisible by 3)

From this, the possible values of \( M \) that make \( 7 + M \) divisible by 3 are \( M = 2, 5, 8 \).

### Step 3: Check divisibility by 4
Next, we need to ensure that \( 34M \) is also divisible by 4. A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

The last two digits of \( 34M \) are \( 4M \). We will check each of the possible values of \( M \):

- If \( M = 2 \): The last two digits are \( 42 \). \( 42 \div 4 = 10.5 \) (not divisible by 4)
- If \( M = 5 \): The last two digits are \( 45 \). \( 45 \div 4 = 11.25 \) (not divisible by 4)
- If \( M = 8 \): The last two digits are \( 48 \). \( 48 \div 4 = 12 \) (divisible by 4)

### Conclusion
The only value of \( M \) that makes \( 34M \) divisible by both 3 and 4 is \( M = 8 \).

Thus, the value of \( M \) is **8**.

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