AMC 10 · 2020 · #6

Grade 4 number-theory

Archetype Small-Domain Constrained Search

digit-constraintsdivisibility-rulessystematic-enumerationmultiplesparity caseworksystematic-enumeration ↑ Prerequisites: divisibility-rulesdigit-constraints

📏 Short solution 💡 2 insights

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Problem

How many $4$ -digit positive integers (that is, integers between $1000$ and $9999$ , inclusive) having only even digits are divisible by $5?$

$\textbf{(A) } 80 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 125 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 500$

Pick an answer.

(A)

(B)

100

(C)

125

(D)

200

(E)

500

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Toolkit + CCSS Solution

Understand

Restated: Count the 4-digit whole numbers (from $1000$ to $9999$) whose every digit is even AND that are divisible by $5$.

Givens: Range: $1000 \le n \le 9999$ (exactly $4$ digits); Every one of the $4$ digits is even: each $\in \{0, 2, 4, 6, 8\}$; $n$ is divisible by $5$; Choices: (A) $80$, (B) $100$, (C) $125$, (D) $200$, (E) $500$

Unknowns: The total count of such numbers

Understand

Restated: Count the 4-digit whole numbers (from $1000$ to $9999$) whose every digit is even AND that are divisible by $5$.

Plan

Primary tool: #2 Make a Systematic List

Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities

The four digits are independent once we pin down their allowed sets — Tool #7 splits the count into four subproblems (one per digit position). Tool #2 lists the allowed values for each position so we don't miss or double-count. Tool #3 eliminates illegal values: odd digits at every position, and $0$ at the thousands place. Multiplying the four counts gives the answer in one line.

Execute — Answer: B

#3 Eliminate Possibilities 4.OA.B.4 Step 1

Force the units digit.
Divisible by $5$ means units is $0$ or $5$.
Even means units is $0$, $2$, $4$, $6$, or $8$.
The only digit in both lists is $0$.

$$\text{units digit} = 0\quad(1\text{ choice})$$

💡 Two constraints overlap to leave a single legal digit.

#2 Make a Systematic List 2.NBT.A.1 Step 2

List the legal thousands digits.
Even digits are $\{0, 2, 4, 6, 8\}$ — but the leading digit cannot be $0$, so cross it out.

$$\text{thousands} \in \{2, 4, 6, 8\}\quad(4\text{ choices})$$

💡 A $4$-digit number must start with a non-zero digit.

#2 Make a Systematic List 2.NBT.A.1 Step 3

List the legal hundreds digits.
Just 'even', no other restriction.

$$\text{hundreds} \in \{0, 2, 4, 6, 8\}\quad(5\text{ choices})$$

💡 Any even digit works in a non-leading slot.

#2 Make a Systematic List 2.NBT.A.1 Step 4

Same list for the tens digit.

$$\text{tens} \in \{0, 2, 4, 6, 8\}\quad(5\text{ choices})$$

💡 Tens place follows the same even rule, no leading constraint.

#7 Identify Subproblems 3.OA.A.1 Step 5

Each digit is chosen independently, so the total count is the product of the four counts (multiplication principle).

$$4 \times 5 \times 5 \times 1 = 100$$

💡 Independent choices multiply — like a $4$-slot decision tree.

#3 Eliminate Possibilities 3.OA.A.3 Step 6

$100$ matches choice (B).

$$N = 100\;\Rightarrow\;\textbf{(B)}$$

💡 Read the matching answer choice.

[1] #3 4.OA.B.4 Force the units digit. Divisible by $5$ means units is $0$ or $5$. Even means un

[2] #2 2.NBT.A.1 List the legal thousands digits. Even digits are $\{0, 2, 4, 6, 8\}$ — but the l

[3] #2 2.NBT.A.1 List the legal hundreds digits. Just 'even', no other restriction.

[4] #2 2.NBT.A.1 Same list for the tens digit.

[5] #7 3.OA.A.1 Each digit is chosen independently, so the total count is the product of the fou

[6] #3 3.OA.A.3 $100$ matches choice (B).

Review

Reasonableness: Sanity check the size. Without any constraint there are $9 \cdot 10 \cdot 10 \cdot 10 = 9000$ four-digit numbers. Even-only digits already drop the count by a lot: $4 \cdot 5 \cdot 5 \cdot 5 = 500$ (choice E — but that ignored divisibility by $5$). Adding the divisibility constraint then divides the units-digit options from $5$ down to $1$, giving $500 / 5 = 100$. The answer $100$ fits in the middle of the choices and matches both counting paths.

Alternative: Tool #9 (Easier Related Problem): solve the $3$-digit version first — how many $3$-digit numbers (between $100$ and $999$) have all even digits and are divisible by $5$? Hundreds: $4$ choices, tens: $5$, units: $1$. Total $20$. The pattern then generalizes — adding one more even digit slot multiplies by $5$, giving $20 \cdot 5 = 100$ for the $4$-digit case.

CCSS standards used (min grade 4)

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite (Using the divisibility rule for $5$ (units digit must be $0$ or $5$) to pin down the units digit.)
2.NBT.A.1 Understand that the three digits of a three-digit number represent hundreds, tens, and ones (Treating a $4$-digit number as four independent place-value slots and listing legal values at each slot.)
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups (Multiplying the four independent slot counts ($4 \times 5 \times 5 \times 1$) to get the total.)
3.OA.A.3 Solve multiplication and division word problems within 100 (Computing $4 \cdot 5 \cdot 5 = 100$ and matching it to the answer choice.)

⭐ This AMC 10 problem only needs Grade 4 'multiples of $5$ end in $0$ or $5$' plus place value you already know — units must be $0$, leading digit picks from $\{2, 4, 6, 8\}$, middle two pick from all five even digits, so $4 \times 5 \times 5 \times 1 = 100$.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 4)

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