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

Introduction / Context:
This question asks for a count of numbers in a given range that are divisible by 2, 3 and 7 at the same time. When a number is required to be divisible by several integers simultaneously, we use the concept of the least common multiple. This problem tests understanding of least common multiples, working with ranges and counting multiples correctly.

Given Data / Assumptions:

- The range of interest is from 300 to 700 inclusive. - Numbers must be divisible by 2, by 3 and by 7 at the same time. - All three divisibility constraints must hold for the same number.

Concept / Approach:
A number divisible by 2, 3 and 7 simultaneously must be divisible by their least common multiple. Since 2, 3 and 7 are pairwise coprime, their least common multiple is the product 2 * 3 * 7 = 42. Therefore, the question reduces to counting the multiples of 42 between 300 and 700 inclusive. Once we identify the first multiple of 42 at or above 300 and the last multiple at or below 700, we can use a simple counting formula.

Step-by-Step Solution:
Step 1: Compute the least common multiple of 2, 3 and 7. Step 2: Since 2, 3 and 7 are coprime, lcm(2, 3, 7) = 2 * 3 * 7 = 42. Step 3: Find the smallest multiple of 42 that is greater than or equal to 300. Compute 300 ÷ 42. Step 4: 42 * 7 = 294 (less than 300), and 42 * 8 = 336 (at least 300). So the first relevant multiple is 336. Step 5: Find the largest multiple of 42 that is less than or equal to 700. Compute 700 ÷ 42. Step 6: 42 * 16 = 672 and 42 * 17 = 714 (greater than 700). So the last relevant multiple is 672. Step 7: The multiples of 42 from 336 to 672 are 336, 378, 420, 462, 504, 546, 588, 630 and 672. Step 8: Count these terms. There are 9 multiples in total.

Verification / Alternative check:
We can verify the count by using a formula. The general k-th multiple of 42 is 42k. We found that k runs from 8 to 16. The number of integers from 8 to 16 inclusive is 16 − 8 + 1 = 9. This matches our manual list and confirms that there are exactly nine numbers in the given range divisible by 2, 3 and 7 simultaneously.

Why Other Options Are Wrong:
Options 7, 11 and 5 do not match the actual count of multiples. If we mistakenly started at 294 or ended at 714, we would miscount the valid numbers. Careful identification of the first and last multiples within the range is essential to avoid such errors.

Common Pitfalls:
A frequent mistake is to count numbers divisible by 2, by 3 and by 7 separately and then try to combine counts, which leads to overcounting or undercounting. Another mistake is to treat 2, 3 and 7 independently rather than using the least common multiple. Using lcm simplifies the problem greatly and provides a straightforward counting method.

Final Answer:
The number of integers between 300 and 700 that are divisible by 2, 3 and 7 simultaneously is 9, which corresponds to option B.