How many positive integers less than 200 are multiples of both 5 and 6?

Introduction / Context: This question checks understanding of multiples and the least common multiple (LCM). We are asked to count how many numbers below 200 are multiples of both 5 and 6. A number that is a multiple of both 5 and 6 must be a multiple of their LCM. This is a standard idea in number systems and divisibility.

Given Data / Assumptions:

- We consider positive integers less than 200.

- Each required number must be divisible by both 5 and 6.

- We will use the least common multiple of 5 and 6 to find such numbers.

Concept / Approach: To be a multiple of both 5 and 6, a number must be a multiple of LCM(5, 6). Once we find the LCM, we can list or count its multiples that are strictly less than 200. This avoids checking divisibility by 5 and 6 separately for every number.

Step-by-Step Solution: Step 1: Find LCM of 5 and 6. Prime factorisation: 5 = 5, 6 = 2 * 3. So LCM(5, 6) = 2 * 3 * 5 = 30. Step 2: List multiples of 30 that are less than 200. They are 30, 60, 90, 120, 150, 180. Step 3: Count these multiples. There are 6 such numbers.

Verification / Alternative check: Check one example, say 90: 90 / 5 = 18 and 90 / 6 = 15, so it is divisible by both. Similarly, 180 / 5 = 36 and 180 / 6 = 30. All listed numbers are clearly multiples of 30 and hence of 5 and 6. No other multiples of 30 less than 200 are missing.

Why Other Options Are Wrong: - 21: Overestimates the count; there are not so many multiples of 30 below 200. - 7: Would imply 7 multiples, but 30 * 7 = 210, which is not less than 200. - 9: Also too large, since 9 multiples would reach 270 which exceeds 200.

Common Pitfalls: Some learners mistakenly add or average 5 and 6 instead of using LCM. Others include 0 or include 200 itself despite the wording “less than 200”. Misinterpreting “both 5 and 6” as “either 5 or 6” is another typical error.

Final Answer: The number of integers less than 200 that are multiples of both 5 and 6 is 6.