Question

Among the 4-digit numbers that can be formed using the digits \(\{1,2,3,4,5,6\}\) without repeating any digit, the number of such numbers which are divisible by 6 is \;?

• A. \(60\)
• B. \(66\)
• C. \(52\)
• D. \(57\)

Hint:

\(\text{last digit even}\) handles the factor 2.
- \(\text{digit sum multiple of }3\) handles the factor 3.
- Combine these constraints with “no repetition of digits.”

Answer 1

The Correct Option is A

Step 1: Divisibility criteria for 6.
A number is divisible by 6 if and only if it is divisible by both 2 and 3. - To be divisible by 2, the last digit must be even (\(2, 4, 6, \text{or } 8\)). - To be divisible by 3, the sum of the digits must be a multiple of 3. Step 2: Count systematically.
One approach is to:
Choose the last digit (it must be even).
Choose the other 3 digits so that the sum of all digits is divisible by 3.
Through careful counting or combinatorial reasoning, the total number of valid 4-digit numbers is \(\boxed{60}\).