Question

All possible 3-digit numbers are formed using the digits \(0,2,3,5,7,9\) without repeating any digit. Then the number of numbers among them which are divisible by \(15\) is

• A. \(11\)
• B. \(14\)
• C. \(16\)
• D. \(36\) \bigskip

Hint:

For divisibility by \(15\), first fix the last digit (\(0\) or \(5\)), then use the divisibility rule for \(3\). Counting by cases is usually the fastest approach in such permutation problems.

Answer 1

The Correct Option is A

Concept: A number is divisible by \(15\) if and only if it is divisible by both \(3\) and \(5\). Therefore,
• The last digit must be \(0\) or \(5\).
• The sum of its digits must be divisible by \(3\). We count the required numbers case by case.

Step 1: Case I: Numbers ending in \(0\). Let the number be \[ \overline{ab0}. \] For divisibility by \(3\), \[ a+b+0 \] must be divisible by \(3\). The available non-zero digits are \[ 2,3,5,7,9. \] Their remainders modulo \(3\) are \[ 2,0,2,1,0. \] The pairs whose sum is divisible by \(3\) are: \[ (3,9),\quad (2,7),\quad (5,7). \] For each pair, the two digits can be arranged in \[ 2! \] ways in the hundred's and ten's places. Hence, the number of required numbers is \[ 3\times 2=6. \]

Step 2: Case II: Numbers ending in \(5\). Let the number be \[ \overline{ab5}. \] For divisibility by \(3\), \[ a+b+5 \] must be divisible by \(3\). Since \[ 5\equiv 2 \pmod 3, \] we require \[ a+b\equiv 1 \pmod 3. \] The available digits are \[ 0,2,3,7,9. \] Their remainders modulo \(3\) are \[ 0,2,0,1,0. \] To obtain remainder \(1\), choose one digit with remainder \(1\) and one with remainder \(0\). The digit with remainder \(1\) is \[ 7. \] The digits with remainder \(0\) are \[ 0,3,9. \] Thus the possible pairs are \[ (7,0),\quad (7,3),\quad (7,9). \] Now count arrangements: \[ (7,0): \] The arrangement \(075\) is not a 3-digit number, so only \[ 705 \] is valid. Thus, \(1\) number. \[ (7,3): \] Both arrangements \[ 735,\;375 \] are valid. Thus, \(2\) numbers. \[ (7,9): \] Both arrangements \[ 795,\;975 \] are valid. Thus, \(2\) numbers. Hence total numbers in this case are \[ 1+2+2=5. \]

Step 3: Find the total count. Therefore, \[ \text{Total} = 6+5 = 11. \]

Step 4: Write the final answer. \[ \boxed{11} \]