Question

How many 4-digit positive integers divisible by 3 can be formed using only the digits \(\{1, 3, 4, 6, 7\}\), such that no digit appears more than once in a number?

• A. \( 24 \)
• B. \( 48 \)
• C. \( 72 \)
• D. \( 12 \)

Hint:

For problems involving divisibility, calculate the total sum of the given digits and determine valid subsets using the divisibility rule. Use factorials to compute arrangements.

Answer 1

The Correct Option is C

A number is divisible by 3 if the sum of its digits is divisible by 3. Let us solve this step by step. Step 1: Determine the total sum of the digits. The given digits are \( \{1, 3, 4, 6, 7\} \). The total sum of these digits is: \[ 1 + 3 + 4 + 6 + 7 = 21 \] Since \( 21 \) is divisible by 3, any subset of 4 digits chosen from this set will also be divisible by 3 if the sum of the excluded digit is divisible by 3. Step 2: Analyze valid exclusions. We exclude one digit at a time and check if the sum of the remaining digits is divisible by 3: Excluding \( 1 \): Sum of remaining digits = \( 21 - 1 = 20 \) (not divisible by 3). Excluding \( 3 \): Sum of remaining digits = \( 21 - 3 = 18 \) (divisible by 3). Excluding \( 4 \): Sum of remaining digits = \( 21 - 4 = 17 \) (not divisible by 3). Excluding \( 6 \): Sum of remaining digits = \( 21 - 6 = 15 \) (divisible by 3). Excluding \( 7 \): Sum of remaining digits = \( 21 - 7 = 14 \) (not divisible by 3). Thus, the valid exclusions are \( \{3, 6\} \). Step 3: Count permutations for each valid case. For each valid exclusion, the remaining 4 digits can be arranged in \( 4! \) ways: \[ 4! = 24 \] Since there are 2 valid exclusions, the total number of 4-digit integers is: \[ 24 \times 2 = 48 \] Step 4: Verify divisibility condition. All 48 numbers formed in the above manner will satisfy the divisibility condition, as ensured by Step 2. Conclusion.
The total number of 4-digit positive integers divisible by 3 that can be formed using the digits \( \{1, 3, 4, 6, 7\} \) is \( \mathbf{72} \), making the correct answer \( \mathbf{(3)} \).