AMC 8 · 2017 · #8
Grade 4 number-theorylogicProblem
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."
(1) It is prime.
(2) It is even.
(3) It is divisible by 7.
(4) One of its digits is 9.
This information allows Malcolm to determine Isabella's house number. What is its units digit?
Pick an answer.
AMC 8 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: 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.
Givens: The house number has two digits ($10$ through $99$); Exactly three of the four statements are true and exactly one is false; Statement (1): the number is prime; Statement (2): the number is even; Statement (3): the number is divisible by $7$; Statement (4): one of the digits is $9$; Answer choices for the units digit: (A) $4$, (B) $6$, (C) $7$, (D) $8$, (E) $9$
Unknowns: The units digit of Isabella's house number
Understand
Restated: 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.
Givens: The house number has two digits ($10$ through $99$); Exactly three of the four statements are true and exactly one is false; Statement (1): the number is prime; Statement (2): the number is even; Statement (3): the number is divisible by $7$; Statement (4): one of the digits is $9$; Answer choices for the units digit: (A) $4$, (B) $6$, (C) $7$, (D) $8$, (E) $9$
Plan
Primary tool: #3 Eliminate Possibilities
Secondary: #2 Make a Systematic List
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 $\mathrm{lcm}(2,7) = 14$ in order and keep only the one whose digits include a $9$.
Execute — Answer: D
4.OA.B.4 Step 1 - Look for contradictions among the four statements when the number has two digits.
- If (1) "prime" and (2) "even" were both true, the number would have to be $2$ (the only even prime), but $2$ is a one-digit number.
- So (1) and (2) cannot both be true.
💡 Knowing that $2$ is the only even prime is exactly the Grade 4 "prime or composite / factor pairs" skill.
4.OA.B.4 Step 2 - Same kind of clash for statements (1) and (3).
- If the number were prime AND divisible by $7$, the only choice would be $7$ itself, which again has just one digit.
- So (1) and (3) cannot both be true either.
💡 Multiples of $7$ that are also prime is again the Grade 4 "recognize multiples / determine prime" idea.
4.OA.B.4 Step 3 - Use those two clashes to find the false statement.
- If (1) were true, then both (2) and (3) would have to be false — that is two false statements, but the problem allows only one.
- So (1) itself must be the single false statement, and (2), (3), (4) are all true.
💡 Eliminating "(1) is true" leaves "(1) is false" — pure process of elimination on a finite set of cases.
4.OA.B.4 Step 4 - Translate the three surviving true statements into a search.
- The number is even AND divisible by $7$, so it is a multiple of $\mathrm{lcm}(2,7) = 14$.
- List every two-digit multiple of $14$ in order using Tool #2.
💡 Listing the multiples of $14$ within $10$-$99$ is a Grade 4 "recognize multiples" exercise.
1.NBT.B.2 Step 5 - Apply the last true statement: one of the digits must be $9$.
- Scan the list $\{14, 28, 42, 56, 70, 84, 98\}$ — only $98$ contains the digit $9$.
- So the house number is $98$, and its units digit is $8$.
💡 Identifying the units (ones) digit of a two-digit number is the Grade 1 place-value standard.
4.OA.B.4 Look for contradictions among the four statements when the number has two digits 4.OA.B.4 Same kind of clash for statements (1) and (3). If the number were prime AND divi 4.OA.B.4 Use those two clashes to find the false statement. If (1) were true, then both ( 4.OA.B.4 Translate the three surviving true statements into a search. The number is even 1.NBT.B.2 Apply the last true statement: one of the digits must be $9$. Scan the list ${1 Review
Reasonableness: Verify $98$ satisfies exactly three of the four statements. (1) prime? $98 = 2 \times 49$, so NO (false, as required). (2) even? YES. (3) divisible by $7$? $98 = 7 \times 14$, 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).
Alternative: Tool #6 (Guess and Check) on the answer choices: each choice gives a candidate units digit. Combined with statement (3), the number is a multiple of $7$ with that units digit, and combined with statement (4), either the tens digit or the units digit must be $9$. Checking (D) units $= 8$: two-digit multiples of $7$ ending in $8$ are $28$ and $98$; of these $98$ also contains a $9$, and is even — only one number survives, which matches "Malcolm can determine the number uniquely." The other choices either give zero valid numbers or more than one, so (D) is forced.
CCSS standards used (min grade 4)
4.OA.B.4Find all factor pairs and recognize multiples; determine prime or composite (Knowing that $2$ is the only even prime and $7$ the only prime multiple of $7$ (to find the contradictions), and listing the two-digit multiples of $14$ as the candidate pool.)1.NBT.B.2Understand that the two digits of a two-digit number represent tens and ones (Reading off the units (ones) digit of $98$ as $8$ to answer the question.)
⭐ This AMC 8 problem only needs Grade 4 ideas about primes and multiples you already know!
⭐ This AMC 8 problem only needs Grade 4 ideas about primes and multiples you already know!
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