AMC 8 · 2017 · #8

Grade 4 number-theorylogic

Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration

prime-numbersmultiplesparitydigit-constraintslcm caseworksystematic-enumerationlogical-deduction ↑ Prerequisites: multiplesparityprimality-test

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Problem

Isabella's house number is a two-digit number. Of the four statements about it — (1) it is prime, (2) it is even, (3) it is divisible by 7, (4) one of its digits is 9 — exactly three are true and one is false. Use that to pin down the unique house number, then report its units digit.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Eliminate Possibilities

We are told exactly one of four statements is false. The cleanest way in is Tool #3 (Eliminate Possibilities): test each statement as the candidate "false" one and rule out the cases that lead to a contradiction. Statement (1) "prime" clashes with (2) "even" (only the prime 2 is even, but 2 has one digit) and with (3) "divisible by 7" (only the prime 7 is divisible by 7, but 7 has one digit). So (1) has to be the false statement, forcing (2), (3), (4) to be true. From there Tool #2 (Systematic List) takes over: list every two-digit multiple of lcm(2,7) = 14 in order and keep only the one whose digits include a 9.

1STEP 1

A two-digit number that is both prime and even would have to be 2, a single digit — so (1) and (2) can't both be true.

prime ∩ even = {2}, and 2 has 1 digit

2STEP 2

Likewise a prime that is also a multiple of 7 must be 7 itself, one digit — so (1) and (3) can't both be true.

prime ∩ multiples of 7 = {7}, and 7 has 1 digit

3STEP 3

If (1) were true, (2) and (3) would both fail — two falses, not allowed — so (1) is the false statement and (2), (3), (4) are true.

(1) true → (2),(3) both false — contradicts "exactly one false"

4STEP 4

Even and divisible by 7 means a multiple of lcm(2,7) = 14, so list the two-digit multiples: 14, 28, 42, 56, 70, 84, 98.

14, 28, 42, 56, 70, 84, 98

5STEP 5

Only 98 in that list has a digit 9, so the house number is 98 and its units digit is 8 — choice (D).

98 → units digit = 8 → (D)

Answer

Verify 98 satisfies exactly three of the four statements. (1) prime? 98 = 2 × 49, so NO (false, as required). (2) even? YES. (3) divisible by 7? 98 = 7 × 14, YES. (4) one digit is 9? YES (the tens digit). Exactly one false and three true — consistent. The units digit 8 matches answer choice (D).

💡Key takeaway

This AMC 8 problem only needs Grade 4 ideas about primes and multiples you already know!

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