Problem 12 · 2003 AMC 8
Medium
Counting & Probability
divisibilitycasework
When a fair six-sided die is tossed on a table top, the bottom face cannot be seen. What is the probability that the product of the numbers on the five faces that can be seen is divisible by 6?
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Answer: E — 1 (it always happens).
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Hint 1 of 2
Don't multiply anything — 6 = 2 × 3, so you only need a 2 and a 3 to BOTH be among the faces you can see.
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Hint 2 of 2
Only one face hides at a time. Ask the worst case: even if the hidden face is the 6, is a 2 and a 3 still showing?
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Approach: prove the event is certain (probability 1)
- A product is divisible by 6 the moment it contains a factor of 2 and a factor of 3 (since 6 = 2 × 3). So we just need both a 2 and a 3 visible — no multiplying required.
- Test the worst case: only one face hides. If the hidden face is the 6, the 2 and 3 are still both up, so the product has 2 × 3. If the hidden face is anything else, then the 6 itself is showing, which already supplies the factor of 6.
- There is no roll that fails, so the event always happens — probability 1.
- You'll see this again: when every outcome works, the probability is exactly 1 (a certain event). Checking the single worst case is enough to prove "always."
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