From the integers 1 to 100 inclusive, how many numbers are divisible by 5 and also contain the digit 5 at least once?

Introduction / Context:This counting problem combines divisibility with a digit-presence condition. Numbers divisible by 5 end in 0 or 5; among them, we must count those that also have the digit 5 somewhere in their decimal representation.

Given Data / Assumptions:

• Range: 1 to 100 inclusive.
• Divisible by 5 ⇒ last digit is 0 or 5.
• Digit 5 must appear in at least one position.

Concept / Approach:Partition divisible-by-5 numbers into those ending with 5 and those ending with 0. The first group automatically contains a 5. The second group contains a 5 only if the tens digit is 5 (i.e., the number 50 within 1–100).

Step-by-Step Solution:

Verification / Alternative check:Scan all multiples of 5 from 5 to 100; the only candidate ending with 0 that contains a 5 is 50. Count remains 11.

Why Other Options Are Wrong:10 ignores 50; 12 overcounts by adding a non-qualifying multiple; 20 counts all multiples of 5 regardless of the digit condition; 9 misses more than one qualifying number.

Common Pitfalls:Forgetting to include 50 or accidentally including 100 (which has no digit 5).

Final Answer:11