How many integers between 300 and 650 inclusive are divisible by both 5 and 7 (that is, divisible by 35)?

Introduction / Context:
This problem checks understanding of divisibility and counting of multiples within a closed interval. Specifically, we are asked to find numbers that are divisible by both 5 and 7, which can be reduced to counting multiples of their least common multiple. Such questions occur frequently in quantitative aptitude tests and also reinforce the concept of least common multiple.

Given Data / Assumptions:
- We are considering all integers from 300 to 650 inclusive. - We want numbers that are divisible by 5 and by 7. - There is no other restriction such as positivity, but the interval is entirely positive anyway.

Concept / Approach:
- A number divisible by both 5 and 7 is divisible by the least common multiple of 5 and 7. - Since 5 and 7 are prime, their least common multiple is 5 * 7 = 35. - The task reduces to counting multiples of 35 between 300 and 650 inclusive. - Use the formula for counting multiples in an interval using integer division.

Step-by-Step Solution:
Step 1: Compute the least common multiple of 5 and 7. Since they are prime and share no common factors, lcm(5, 7) = 5 * 7 = 35. Step 2: Find the smallest multiple of 35 that is greater than or equal to 300. Divide 300 by 35 to get 300 / 35 which is approximately 8.57. Step 3: The smallest integer k such that 35k is at least 300 is k = 9. Then 35 * 9 = 315, which is the first valid multiple. Step 4: Find the largest multiple of 35 that is less than or equal to 650. Compute 650 / 35 which is approximately 18.57. Step 5: The largest integer k with 35k at most 650 is k = 18. Then 35 * 18 = 630, which is the last valid multiple. Step 6: Now count how many integers k from 9 to 18 inclusive there are. This is 18 - 9 + 1 = 10. Step 7: Therefore, there are 10 numbers between 300 and 650 inclusive that are divisible by both 5 and 7.

Verification / Alternative check:
List the multiples of 35 from 315 to 630 to verify: 315, 350, 385, 420, 455, 490, 525, 560, 595, 630. That is exactly 10 numbers. Each is clearly divisible by 5, as they end in 0 or 5, and also divisible by 7, since they are multiples of 35. This confirms our count.

Why Other Options Are Wrong:
Option A (8): This undercounts the multiples. There are more than 8 multiples of 35 in the given range. Option B (9): This still misses one valid multiple, so it is too small. Option D (12): This overcounts and would require extending beyond 630 or including numbers below 315. Option E (11): This is also too large; the explicit list confirms there are only 10.

Common Pitfalls:
- Forgetting to use the least common multiple and instead trying to filter by 5 and 7 separately without a systematic method. - Off by one errors when counting endpoints, especially when using floor and ceiling division. - Including numbers like 300 or 650 without checking divisibility carefully.

Final Answer:
The number of integers between 300 and 650 that are divisible by both 5 and 7 is 10.