Between 1 and 200 inclusive, how many integers are divisible by 3 but not divisible by 7?

Introduction / Context:
This counting problem tests inclusion exclusion logic for divisibility. We need to count numbers in an interval that satisfy one divisibility condition (divisible by 3) but explicitly fail another (not divisible by 7). Such problems help build the habit of using least common multiple and simple counting formulas systematically.

Given Data / Assumptions:
- We consider all integers from 1 to 200 inclusive. - We want numbers that are divisible by 3. - Among those, we must exclude numbers that are also divisible by 7. - The answer should be a simple count of such integers.

Concept / Approach:
- First count how many numbers between 1 and 200 are divisible by 3. - Then count how many numbers between 1 and 200 are divisible by both 3 and 7, that is, by lcm(3, 7) = 21. - Subtract the count of numbers divisible by 21 from the count of numbers divisible by 3 to get the final answer.

Step-by-Step Solution:
Step 1: Count multiples of 3 between 1 and 200. The smallest multiple is 3 and the largest multiple less than or equal to 200 is 198. Step 2: The number of multiples of 3 is 200 divided by 3, taking the integer part. 200 / 3 is approximately 66.66, so there are 66 multiples of 3. Step 3: Now find numbers that are multiples of both 3 and 7. The least common multiple of 3 and 7 is 21. Step 4: Count multiples of 21 between 1 and 200. The smallest multiple is 21 and the largest is 189. Step 5: The number of multiples of 21 is 200 divided by 21, taking the integer part. 200 / 21 is approximately 9.52, so there are 9 such multiples. Step 6: All numbers divisible by both 3 and 7 are also divisible by 21, so we must exclude these 9 from the 66 multiples of 3. Step 7: Therefore, the count of numbers divisible by 3 but not by 7 is 66 - 9 = 57.

Verification / Alternative check:
As a check, imagine listing multiples of 3 up to 200: 3, 6, 9, ..., 198. Among these, every seventh multiple of 3 starting from 21 (that is, 21, 42, 63, 84, 105, 126, 147, 168, 189) is also divisible by 7. This list clearly has 9 terms, confirming that 9 multiples must be excluded, and thus 57 remain.

Why Other Options Are Wrong:
Option A (38): This is too small and would correspond to miscounting the multiples or using the wrong interval. Option B (47): This also undercounts, ignoring some valid multiples of 3. Option D (66): This counts all multiples of 3 and fails to remove those that are also divisible by 7. Option E (60): This does not match the precise inclusion exclusion calculation and is not backed by a consistent method.

Common Pitfalls:
- Forgetting to subtract the numbers that are divisible by both 3 and 7, leading to an overcount. - Miscalculating the floor of 200 / 3 or 200 / 21. - Confusing the interval boundaries (for example, accidentally using up to 201 instead of 200).

Final Answer:
There are 57 integers between 1 and 200 that are divisible by 3 but not by 7.