Problem 5 · 2024 AMC 8
Stretch
Number Theory
divisibilityfactor-pairscasework
Aaliyah rolls two standard 6-sided dice. She notices that the product of the two numbers rolled is a multiple of 6. Which of the following integers cannot be the sum of the two numbers?
Show answer
Answer: B — The sum cannot be 6.
Show hints
Hint 1 of 2
What does "multiple of 6" really demand of the two dice? Break 6 into its prime pieces — the product needs both of them.
Still stuck? Show hint 2 →
Hint 2 of 2
Technique: 6 = 2 × 3, so the pair needs a factor of 2 (an even die) AND a factor of 3 (a 3 or 6). Test each answer's possible pairs against that single rule.
Show solution
Approach: test which sum has no multiple-of-6 pair
- The insight: "multiple of 6" means "has a factor 2 AND a factor 3." So the pair must contain at least one even number AND at least one 3 or 6. That's the only condition — no need to multiply anything out.
- Now test each sum. Sum 6 can only be (1,5), (2,4), or (3,3). Check: (1,5) has no even-and-3, (2,4) has no 3 or 6, (3,3) has a 3 but no even number. None qualify — so a sum of 6 is impossible.
- Every other choice does have a qualifying pair: 5 = (2,3), 7 = (1,6), 8 = (2,6), 9 = (3,6). Answer: 6. This transfers: to test divisibility by a composite like 6, 12, or 15, split it into prime factors and check each one separately.
Another way — list every valid pair (MAA):
- Pairs whose product is a multiple of 6 (need a multiple of 3 and an even number): (1,6), (2,3), (2,6), (3,6), (4,6), (5,6), (6,6).
- Their sums: 7, 5, 8, 9, 10, 11, 12. Among A–E, only 6 is missing.
Mark:
· log in to save