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:

For problems involving multiples, find the least common multiple (LCM) to determine which numbers must divide the product.

Answer 1

Answer: (E)

Step 1: Understand the multiples.
We are given that:
- \( x \) is a multiple of 4, so \( x = 4k \) for some integer \( k \).
- \( y \) is a multiple of 6, so \( y = 6m \) for some integer \( m \).
Thus, the product \( xy \) is: \[ xy = (4k)(6m) = 24km \] This shows that \( xy \) is always a multiple of 24, which is divisible by 8, 12, and 18.
Step 2: Check each condition.
- \( xy \) is divisible by 8, because \( 24 \) is divisible by 8.
- \( xy \) is divisible by 12, because \( 24 \) is divisible by 12.
- \( xy \) is divisible by 18, because \( 24 \) is divisible by 18.
Thus, \( xy \) is a multiple of 8, 12, and 18. \[ \boxed{I, II, \text{and} III} \]