To determine if the number \( 24x \) is divisible by 6, we need to check two conditions: whether it is divisible by 2 and whether it is divisible by 3.
### Step 1: Check divisibility by 2
A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8). In our case, the last digit of \( 24x \) is \( x \).
- If \( x \) is 0, 2, 4, 6, or 8, then \( 24x \) is divisible by 2.
- If \( x \) is 1, 3, 5, 7, or 9, then \( 24x \) is not divisible by 2.
### Step 2: Check divisibility by 3
A number is divisible by 3 if the sum of its digits is divisible by 3. The digits of \( 24x \) are 2, 4, and \( x \).
- The sum of the digits is \( 2 + 4 + x = 6 + x \).
- Now we need to check if \( 6 + x \) is divisible by 3.
To check this:
- If \( x = 0 \), then \( 6 + 0 = 6 \) (divisible by 3).
- If \( x = 1 \), then \( 6 + 1 = 7 \) (not divisible by 3).
- If \( x = 2 \), then \( 6 + 2 = 8 \) (not divisible by 3).
- If \( x = 3 \), then \( 6 + 3 = 9 \) (divisible by 3).
- If \( x = 4 \), then \( 6 + 4 = 10 \) (not divisible by 3).
- If \( x = 5 \), then \( 6 + 5 = 11 \) (not divisible by 3).
- If \( x = 6 \), then \( 6 + 6 = 12 \) (divisible by 3).
- If \( x = 7 \), then \( 6 + 7 = 13 \) (not divisible by 3).
- If \( x = 8 \), then \( 6 + 8 = 14 \) (not divisible by 3).
- If \( x = 9 \), then \( 6 + 9 = 15 \) (divisible by 3).
### Step 3: Combine the results
Now we combine the results from both checks:
- For \( 24x \) to be divisible by 6, it must be divisible by both 2 and 3.
- The values of \( x \) that make \( 24x \) divisible by 2 are 0, 2, 4, 6, and 8.
- The values of \( x \) that make \( 6 + x \) divisible by 3 are 0, 3, 6, and 9.
The common values of \( x \) that satisfy both conditions are:
- \( x = 0 \) (divisible by both 2 and 3)
- \( x = 6 \) (divisible by both 2 and 3)
Thus, \( 24x \) is divisible by 6 when \( x = 0 \) or \( x = 6 \).
### Final Answer
The values of \( x \) for which \( 24x \) is divisible by 6 are \( x = 0 \) and \( x = 6 \).
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