Question

If the positive integer x is a multiple of 4 and the positive integer y is a multiple of 6, then xy must be a multiple of which of the following?
I. 8
II. 12
III. 18

• A. II only
• B. I and II only
• C. I and III only
• D. II and III only
• E. I, II, and III

Hint:

To check if a statement about multiples "must be true," try to find a counterexample using the smallest possible values. If you can find one case where the statement fails, it's false. For x being a multiple of 4 and y a multiple of 6, the simplest case is x=4, y=6.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem tests understanding of multiples and factors. We are given properties of two integers, x and y, and asked to determine a necessary property of their product, xy.
Step 2: Key Formula or Approach:
We can express x and y algebraically. Since x is a multiple of 4, we can write \(x = 4a\) for some positive integer a. Since y is a multiple of 6, we can write \(y = 6b\) for some positive integer b. The product is \(xy = (4a)(6b) = 24ab\).
This means that xy must be a multiple of 24. We need to check which of the given options are factors of 24. Alternatively, we can use a counterexample to disprove a statement.
Step 3: Detailed Explanation:
The product \(xy = 24ab\). Let's check each statement. I. 8
Is 24ab always a multiple of 8? Yes, because \(24ab = 8 \times (3ab)\). Since 3ab is an integer, xy is always a multiple of 8. Statement I is true.
II. 12
Is 24ab always a multiple of 12? Yes, because \(24ab = 12 \times (2ab)\). Since 2ab is an integer, xy is always a multiple of 12. Statement II is true.
III. 18
Is 24ab always a multiple of 18? Not necessarily. We can test this with the smallest possible values for x and y. Let \(a=1\) and \(b=1\). Then \(x=4\) and \(y=6\). The product is \(xy = 4 \times 6 = 24\). Is 24 a multiple of 18? No. Since we found a case where xy is not a multiple of 18, this statement is not always true. Statement III is false.
Step 4: Final Answer:
Only statements I and II must be true.