How many integers from 300 to 700 (inclusive) are divisible by 2, 3, and 7?

Introduction / Context:
This question is about counting numbers in a given interval that satisfy multiple divisibility conditions. Here, each required number must be divisible by 2, 3, and 7 simultaneously. This is best solved using the least common multiple (LCM) and counting multiples of that LCM within the range from 300 to 700.

Given Data / Assumptions:

- We consider all integers from 300 to 700 inclusive.

- We want numbers divisible by 2, 3, and 7.

- Divisibility by all three means the number must be a multiple of LCM(2, 3, 7).

Concept / Approach:
First, compute LCM(2, 3, 7). Any number divisible by 2, 3, and 7 must be a multiple of this LCM. Then, find the smallest and largest multiples of this LCM within the interval [300, 700]. Finally, count how many such multiples exist by using a simple arithmetic progression count.

Step-by-Step Solution:
Step 1: Find LCM of 2, 3, and 7. 2 = 2, 3 = 3, 7 = 7. LCM(2, 3, 7) = 2 * 3 * 7 = 42. Step 2: Find the smallest multiple of 42 that is at least 300. 42 * 7 = 294, which is less than 300. 42 * 8 = 336, which lies within the range. So the first valid multiple is 336. Step 3: Find the largest multiple of 42 that is at most 700. 42 * 16 = 672, which is less than or equal to 700. 42 * 17 = 714, which exceeds 700. So the last valid multiple is 672. Step 4: Count the multiples from 336 to 672 inclusive. These are 42 * 8, 42 * 9, ..., 42 * 16. The indices run from 8 to 16 inclusive, giving 16 - 8 + 1 = 9 numbers.

Verification / Alternative check:
List them explicitly to verify: 336, 378, 420, 462, 504, 546, 588, 630, 672. All are divisible by 42, hence by 2, 3, and 7, and all lie between 300 and 700 inclusive. Counting them confirms there are 9 such numbers.

Why Other Options Are Wrong:
- 7 or 8: These underestimate the count; we clearly find 9 valid multiples.
- 10: This would require another multiple of 42 in the range, but 42 * 17 = 714 is outside the interval.

Common Pitfalls:
A common error is to count numbers divisible by 2, by 3, and by 7 separately and try to combine the counts incorrectly. Another mistake is forgetting that the range includes both endpoints (300 and 700), or miscomputing the first or last multiple inside the range. Using LCM with index counting is the most systematic approach.

Final Answer:
There are 9 integers between 300 and 700 that are divisible by 2, 3, and 7.