Question

How many three digit numbers divisible by 5 are there in which no digits are repeated?

• A. 136
• B. 128
• C. 144
• D. 162

Hint:

Whenever the digit 0 is restricted from the first position and is also part of a divisibility condition, always split the counting into two scenarios: "ending in 0" and "ending in the non-zero option".
This prevents errors regarding the placement of 0 in the hundreds place.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A number is divisible by 5 if its last digit is either 0 or 5. Since digits cannot be repeated and the first digit cannot be 0, we must split this into two cases based on the unit's digit.
Step 2: Case 1 - Unit's digit is 0:
If the last digit is 0, the first digit can be any of $\{1, 2, ..., 9\}$ (9 options). The second digit can be any of the remaining 8 digits (since 0 and the first digit are used). Total for Case 1: $9 \times 8 = 72$.
Step 3: Case 2 - Unit's digit is 5:
If the last digit is 5, the first digit can be any of $\{1, 2, 3, 4, 6, 7, 8, 9\}$ (8 options, as it cannot be 0 or 5). The second digit can then be any of the remaining 8 digits (including 0, but excluding the two already used). Total for Case 2: $8 \times 8 = 64$.
Step 4: Final Answer:
Total numbers $= 72 + 64 = 136$.