AMC 10 · 2020 · #6
Grade 4 number-theoryProblem
How many 4-digit positive integers (that is, integers between 1000 and 9999, inclusive) having only even digits are divisible by 5?
Pick an answer.
AMC 10 2020 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: 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 $5$.
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) $500$
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
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 $0$.
💡 Two constraints overlap to leave a single legal digit.
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.
💡 A $4$-digit number must start with a non-zero digit.
2.NBT.A.1 Step 3 - List the legal hundreds digits.
- Just 'even', no other restriction.
💡 Any even digit works in a non-leading slot.
2.NBT.A.1 Step 4 Same list for the tens digit.
💡 Tens place follows the same even rule, no leading constraint.
3.OA.A.1 Step 5 Each digit is chosen independently, so the total count is the product of the four counts (multiplication principle).
💡 Independent choices multiply — like a $4$-slot decision tree.
3.OA.A.3 Step 6 $100$ matches choice (B).
💡 Read the matching answer choice.
4.OA.B.4 Force the units digit. Divisible by $5$ means units is $0$ or $5$. Even means un 2.NBT.A.1 List the legal thousands digits. Even digits are $\{0, 2, 4, 6, 8\}$ — but the l 2.NBT.A.1 List the legal hundreds digits. Just 'even', no other restriction. 2.NBT.A.1 Same list for the tens digit. 3.OA.A.1 Each digit is chosen independently, so the total count is the product of the fou 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.4Find 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.1Understand 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.1Interpret 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.3Solve 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 $4 \times 5 \times 5 \times 1 = 100$.
⭐ 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 $4 \times 5 \times 5 \times 1 = 100$.
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