AMC 8 · 2001 · #4

Grade 4 number-theoryarithmetic

Archetype Small-Domain Constrained Search

place-valuedigit-constraintsparitysystematic-enumeration systematic-enumerationcaseworkdigit-constraints ↑ Prerequisites: place-valueparity

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Problem

The digits 1, 2, 3, 4 and 9 are each used once to form the smallest possible even five-digit number. The digit in the tens place is

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: Use each of the digits $1, 2, 3, 4, 9$ exactly once to build the smallest possible $5$-digit even number. What digit ends up in the tens place?

Givens: The digits $1, 2, 3, 4, 9$ are each used once; The $5$-digit number must be even; The number must be as small as possible; Answer choices: (A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) $9$

Unknowns: The digit in the tens place of the smallest such number

Understand

Restated: Use each of the digits $1, 2, 3, 4, 9$ exactly once to build the smallest possible $5$-digit even number. What digit ends up in the tens place?

Givens: The digits $1, 2, 3, 4, 9$ are each used once; The $5$-digit number must be even; The number must be as small as possible; Answer choices: (A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) $9$

Plan

Primary tool: #6 Guess and Check

Secondary: #5 Look for a Pattern

Tool #5 (Look for a Pattern) gives the place-value rule: in a multi-digit number, the leftmost place matters most, so to keep the number small we should put the smallest available digit on the left first, then the next smallest, and so on. Tool #6 (Guess and Check) handles the twist — the even-number rule forces the units digit to be $2$ or $4$, so the greedy left-to-right plan can run into a conflict when it reaches the last two places. When that happens, we swap the units digit and try again.

Execute — Answer: E

#5 Look for a Pattern 4.NBT.A.2 Step 1

First guess: greedy left-to-right.
Sort the digits $1 < 2 < 3 < 4 < 9$ and fill the places from left to right with the smallest available digit each time.
That gives $12349$.

$$\text{first guess} = 12349$$

💡 Place value says the ten-thousands slot is worth the most, so the smallest digit goes there; then the thousands slot, and so on.

#6 Guess and Check 2.OA.C.3 Step 2

Check the even-number rule.
The units digit of $12349$ is $9$, which is odd.
So the guess fails.
We need the units digit to be $2$ or $4$ — the only even digits in the set.

$$12349 \text{ ends in } 9 \Rightarrow \text{odd} \Rightarrow \text{not allowed}$$

💡 A number is even exactly when its units digit is even, and the only even digits available are $2$ and $4$.

#6 Guess and Check 4.NBT.A.2 Step 3

Fix the units digit, then reapply the place-value rule.
To keep the number as small as possible, leave the units digit $9$ for last and pick the larger even digit, $4$, to sit in the units place.
That frees up the small digit $2$ to sit further left, where it lowers the number more.
Now fill the remaining four places with $1, 2, 3, 9$ from left to right, smallest first: $1, 2, 3$ go in the ten-thousands, thousands, hundreds places, and $9$ is left over for the tens place.

$$\underbrace{1}_{\text{ten-th}}\ \underbrace{2}_{\text{th}}\ \underbrace{3}_{\text{hun}}\ \underbrace{9}_{\text{tens}}\ \underbrace{4}_{\text{units}} = 12394$$

💡 Sending $4$ (rather than $2$) to the units place is better because $2$ is smaller and helps more in the higher places.

#5 Look for a Pattern 4.NBT.A.2 Step 4

Read off the tens digit of $12394$.

$$12394 \Rightarrow \text{tens digit} = 9 \Rightarrow \textbf{(E)}$$

💡 The tens place is the second digit from the right, which is $9$.

[1] #5 4.NBT.A.2 First guess: greedy left-to-right. Sort the digits $1 < 2 < 3 < 4 < 9$ and fill

[2] #6 2.OA.C.3 Check the even-number rule. The units digit of $12349$ is $9$, which is odd. So

[3] #6 4.NBT.A.2 Fix the units digit, then reapply the place-value rule. To keep the number as sm

[4] #5 4.NBT.A.2 Read off the tens digit of $12394$.

Review

Reasonableness: Compare against the other legal candidate. If we had put $2$ in the units place instead, the remaining digits $1, 3, 4, 9$ filled left-to-right give $13492$. Since $12394 < 13492$, the choice of $4$ in the units place was correct. Also check $12394$ uses each of $1, 2, 3, 4, 9$ exactly once (yes) and ends in $4$ (even, yes). Finally, the tens digit is the second from the right: $1\,2\,3\,\underline{9}\,4 \to 9$, matching answer (E).

Alternative: Tool #2 (Make a Systematic List): the units digit must be $2$ or $4$. For each choice, list the smallest number formed by the remaining digits in increasing order from the left. Units $= 2$ leaves $\{1,3,4,9\}$, giving $13492$. Units $= 4$ leaves $\{1,2,3,9\}$, giving $12394$. The smaller is $12394$, whose tens digit is $9$.

CCSS standards used (min grade 4)

4.NBT.A.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form (Using place value to know that the leftmost digit has the largest weight, so the smallest available digit should be placed there to make the overall number as small as possible.)
2.OA.C.3 Determine whether a group of objects (up to 20) has an odd or even number of members (Recognizing that a whole number is even exactly when its units digit is even, which forces the units digit of the answer to be $2$ or $4$.)

⭐ To shrink a number, push the smallest digits to the leftmost places — but if a rule like "must be even" claims a spot, give that spot the larger acceptable digit so the smaller ones can do more work on the left.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 4)

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