Between 300 and 700 (inclusive), how many integers are divisible simultaneously by 2, 3 and 7, that is, by their common multiple?

Introduction:
This divisibility and counting question asks you to determine how many numbers in a given interval are divisible by three given integers simultaneously: 2, 3 and 7. Such problems test your understanding of least common multiple (LCM) and counting multiples in a range.

Given Data / Assumptions:

• We are considering integers from 300 to 700 inclusive.
• Each required integer must be divisible by 2, 3 and 7 at the same time.
• Divisibility by all three means divisibility by their least common multiple.

Concept / Approach:
The LCM of 2, 3 and 7 captures simultaneous divisibility. Once we find this LCM, we count how many multiples of this LCM lie between 300 and 700 inclusive using a simple floor and ceiling method.

Step-by-Step Solution:
Step 1: Compute the LCM of 2, 3 and 7.2, 3 and 7 are pairwise co-prime.LCM = 2 * 3 * 7 = 42.Step 2: Find the first multiple of 42 not less than 300.42 * 7 = 294 (too small).42 * 8 = 336 (within the range, this is the first valid multiple).Step 3: Find the last multiple of 42 not greater than 700.42 * 16 = 672 (within the range).42 * 17 = 714 (greater than 700, so stop at 16).Step 4: Count the multiples from 8 to 16 inclusive.Number of terms = 16 − 8 + 1 = 9.

Verification / Alternative check:
List the actual multiples: 336, 378, 420, 462, 504, 546, 588, 630, 672. There are exactly nine such numbers, all between 300 and 700 inclusive, each divisible by 42 and therefore by 2, 3 and 7.

Why Other Options Are Wrong:
The other counts (7, 11, 5, 8) do not match the actual number of valid multiples. Only 9 corresponds to the correct count obtained from both formula and direct listing.

Common Pitfalls:
Some candidates attempt to count multiples of 2, 3 and 7 separately and then try to intersect them manually, which is inefficient and error-prone. Using the LCM avoids double-counting and simplifies the problem greatly.

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