AMC 8 · 2017 · #8

Easy mode Grade 4

Problem

Malcolm wants to visit Isabella, but he doesn't know her house number. Isabella gives him a clue.

She tells him: "My house number is a two-digit number. Look at the four statements below. Exactly three of them are true, and one of them is false."

(1) It is prime.

(2) It is even.

(3) It is divisible by $7$ .

(4) One of its digits is $9$ .

This clue is enough for Malcolm to figure out the exact house number. So the four statements pin down only one possible two-digit number.

What is the ones digit of that house number?

$\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

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

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

#3 Eliminate Possibilities 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.

$\text{prime} \,\cap\, \text{even} = \{2\}$, and $2$ has $1$ digit

💡 Knowing that $2$ is the only even prime is exactly the Grade 4 "prime or composite / factor pairs" skill.

#3 Eliminate Possibilities 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.

$\text{prime} \,\cap\, \text{multiples of }7 = \{7\}$, and $7$ has $1$ digit

💡 Multiples of $7$ that are also prime is again the Grade 4 "recognize multiples / determine prime" idea.

#3 Eliminate Possibilities 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.

$(1) \text{ true} \Rightarrow (2),(3) \text{ both false}$ — contradicts "exactly one false"

💡 Eliminating "(1) is true" leaves "(1) is false" — pure process of elimination on a finite set of cases.

#2 Make a Systematic List 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.

$$14, \; 28, \; 42, \; 56, \; 70, \; 84, \; 98$$

💡 Listing the multiples of $14$ within $10$-$99$ is a Grade 4 "recognize multiples" exercise.

#2 Make a Systematic List 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 $8$.

$$98 \Rightarrow \text{units digit} = 8 \;\Rightarrow\; \textbf{(D)}$$

💡 Identifying the units (ones) digit of a two-digit number is the Grade 1 place-value standard.

[1] #3 4.OA.B.4 Look for contradictions among the four statements when the number has two digits

[2] #3 4.OA.B.4 Same kind of clash for statements (1) and (3). If the number were prime AND divi

[3] #3 4.OA.B.4 Use those two clashes to find the false statement. If (1) were true, then both (

[4] #2 4.OA.B.4 Translate the three surviving true statements into a search. The number is even

[5] #2 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.4 Find 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.2 Understand 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!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 4)

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