Problem 8 · 2017 AMC 8
Medium
Number Theory
divisibilitycasework
Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn't know her house number. She tells him, "My house number has two digits, and exactly three of the following four statements about it are true."
- It is prime.
- It is even.
- It is divisible by 7.
- One of its digits is 9.
This information allows Malcolm to determine Isabella's house number. What is its units digit?
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Answer: D — 8.
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Hint 1 of 2
"Exactly three true" means exactly one is false. Instead of testing numbers, hunt for two statements that fight each other — they can't both be true, so one of them must be the lone false one.
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Hint 2 of 2
A 2-digit number can't be both even and prime (the only even prime is 2). So statement (1) "prime" and (2) "even" clash; the false one must be (1), forcing (2), (3), (4) true.
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Approach: spot the clashing pair, then build the number
- Exactly one statement is false. The pair (1) prime and (2) even can never both hold for a 2-digit number (2 is the only even prime). So the false statement is (1) — that's the deduction that cracks it — and (2), (3), (4) are all true.
- True facts: even AND divisible by 7 means divisible by 14. Two-digit multiples of 14: 14, 28, 42, 56, 70, 84, 98. Only 98 also has a 9 as a digit.
- Its units digit is 8.
- Why this transfers: in "exactly k of these are true" puzzles, look for a forced contradiction first — pinning the false statement collapses the search far faster than checking candidates one by one.
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