Which of the following lists consists only of multiples of 6 and also includes numbers that are common multiples of both 4 and 6?

Introduction:
This question checks your understanding of multiples and common multiples. You must choose a list of numbers that are all multiples of 6, and within that list, some numbers should also be common multiples of both 4 and 6. It is a straightforward exercise in divisibility, especially for small numbers, and reinforces how least common multiples work.

Given Data / Assumptions:

• A multiple of 6 is any number of the form 6k, where k is an integer.
• A common multiple of 4 and 6 must be divisible by both 4 and 6.
• The least common multiple (LCM) of 4 and 6 is 12.
• Any common multiple of 4 and 6 is a multiple of 12.

Concept / Approach:
Check each list in the options. For each number in a list, test if it is divisible by 6. Additionally, see whether at least some of those numbers are divisible by both 4 and 6 (i.e., by 12). The correct option must satisfy both conditions: every number is a multiple of 6, and there are common multiples of 4 and 6 present.

Step-by-Step Solution:
Option A: 12, 18, 36.12/6 = 2, 18/6 = 3, 36/6 = 6, so all are multiples of 6.12 and 36 are also divisible by 4: 12/4 = 3, 36/4 = 9, so they are common multiples of 4 and 6.Option B: 12, 34, 42. Here 34 is not divisible by 6, so this list fails.Option C: 6, 4, 14. Here 4 and 14 are not multiples of 6.Option D: 4, 8, 16. None of these are multiples of 6.Option E: 6, 10, 22. Only 6 is a multiple of 6; 10 and 22 are not.Therefore, only option A satisfies all requirements.

Verification / Alternative check:
Recall that common multiples of 4 and 6 are multiples of 12: 12, 24, 36, 48, and so on. Option A uniquely contains 12 and 36, both of which are on this list, and all entries in option A are multiples of 6. No other option has this combination of properties.

Why Other Options Are Wrong:
Option B: contains 34, which breaks the “multiple of 6” requirement.Option C: mixes 6 with non-multiples 4 and 14.Option D: all numbers are multiples of 4 but not of 6.Option E: includes only one multiple of 6 and two non-multiples.

Common Pitfalls:
Checking only whether a list contains some multiple of 6 rather than verifying all of them.Forgetting that a common multiple of 4 and 6 must be divisible by both, which for small numbers essentially means being a multiple of 12.Confusing multiples with factors.

Final Answer:
12, 18, 36