In number systems, which of the following numbers are multiples of 18?

Introduction:
This problem tests your understanding of multiples of a given integer, here the number 18. Recognizing multiples is an essential skill in arithmetic, divisibility rules, and simplifying ratios.

Given Data / Assumptions:

• The base number is 18.
• The candidate numbers are 72, 144, and 306.
• We must decide which of them are exact multiples of 18.

Concept / Approach:
A number M is a multiple of 18 if there exists an integer k such that: M = 18 * k. Equivalently, M must be divisible by both 2 and 9, since 18 = 2 * 9. However, the most straightforward method here is direct division by 18.

Step-by-Step Solution:
Check 72: 72 / 18 = 4, which is an integer, so 72 is a multiple of 18. Check 144: 144 / 18 = 8, which is an integer, so 144 is a multiple of 18. Check 306: 306 / 18 = 17, which is an integer, so 306 is a multiple of 18. All three given numbers are exact multiples of 18.

Verification / Alternative Check:
We can also factor the numbers: 72 = 8 * 9 = 2^3 * 3^2, which includes 2 * 3^2 = 18 as a factor. 144 = 16 * 9 = 2^4 * 3^2, again containing 18. 306 = 18 * 17 directly. This confirms that each number is divisible by 18.

Why Other Options Are Wrong:
Options 72, 144, and 306 individually are correct as multiples, but the question asks which of the following numbers are multiples. Since all of them are, the best choice is "All of the above." "None of the above" is clearly incorrect because we have already shown that each candidate is a multiple of 18.

Common Pitfalls:
Students sometimes do not divide fully and misjudge divisibility, or they mistakenly think that only larger numbers with certain patterns (like ending in 0) are multiples. Remember that precise division or factorization is the most reliable method.

Final Answer:
All three numbers are multiples of 18, so the correct choice is All of the above.