AMC 10 · 2025 · #8

Grade 6 number-theory

Archetype Small-Domain Constrained Search └ Sub-archetype Factor / Divisibility Constraint

divisibility-rulesdigit-sumdigit-constraints casework ↑ Prerequisites: divisibility-rules

📏 Long solution 💡 3 insights

Problem

Emmy says to Max, "I ordered 36 math club sweatshirts today." Max asks, "How much did each shirt cost?" Emmy responds, "I'll give you a hint. The total cost was $\$ \underline A~\underline B~\underline B.\underline B~\underline A$ , where $A$ and $B$ are digits and $A \neq 0$ ." After a pause, Max says, "That was a good price." What is $A + B$ ?

Pick an answer.

(A)

(B)

(C)

~11

(D)

~14

(E)

~15

AMC 10 2025 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.

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Toolkit + CCSS Solution

Understand

Restated: A total price is written as the money amount $\underline A\,\underline B\,\underline B.\underline B\,\underline A$ dollars, where A and B are digits and A is not 0. This total pays for exactly 36 sweatshirts at a whole-cent price each. Find A + B.

Givens: The total cost has the digit pattern $A\,B\,B.\,B\,A$ (three digits before the decimal point, two after).; A and B are single digits, and $A \neq 0$.; The total is the price of 36 identical sweatshirts, each costing a whole number of cents.

Unknowns: The digits A and B, and therefore the sum $A + B$.

Understand

Plan

Primary tool: #3 Eliminate Possibilities

Secondary: #8 Analyze the Units, #4 Introduce a Variable

The digits A and B can only take a handful of values, so the smart move is to turn each fact into a divisibility test and cross off every pair that fails. First read the price as a whole number of cents so 'divisible by 36' means something exact, then use the rules for 9 and 4 to squeeze the candidates down to a single pair.

Execute — Answer: C

#8 Analyze the Units 5.NBT.A.3 Step 1

Read the price as whole cents

The total is a money amount with the digits A, B, B before the point and B, A after it.
Written as a whole number of cents it is the five-digit number $A\,B\,B\,B\,A$.
Because 36 shirts split the cost evenly and each shirt costs a whole number of cents, this cent total must be divisible by 36.

$$\text{cents} = \underline A\,\underline B\,\underline B\,\underline B\,\underline A,\quad 36 \mid \text{cents}$$

💡 Money in cents is just a plain whole number, so 'splits evenly among 36' becomes a clean divisibility question.

#3 Eliminate Possibilities 4.OA.B.4 Step 2

Split 36 into 4 and 9

A number is divisible by 36 exactly when it is divisible by both 4 and 9, since $36 = 4 \times 9$ and 4 and 9 share no common factor.
So the five-digit number $A\,B\,B\,B\,A$ must satisfy the rule for 9 and the rule for 4 at the same time.

$$36 = 4 \times 9$$

💡 Breaking 36 into 4 and 9 lets you use two easy digit rules instead of dividing a big number.

#4 Introduce a Variable 6.EE.B.6 Step 3

Apply the rule for 9

The rule for 9 says the digit sum must be a multiple of 9.
The digits are A, B, B, B, A, so their sum is $2A + 3B$.
Now $3B$ is already a multiple of 3, so for the whole sum to be a multiple of 9 (and hence of 3), the part $2A$ must also be a multiple of 3.
That forces A to be a multiple of 3.
Since $A \neq 0$, we have $A \in \{3, 6, 9\}$.

$$2A + 3B \equiv 0 \pmod 9 \;\Rightarrow\; 3 \mid A \;\Rightarrow\; A \in \{3,6,9\}$$

💡 Since 3B is already a multiple of 3, A has to carry the rest of the load and be a multiple of 3 too.

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

Use divisibility by 4 to pin A

A number divisible by 4 is even, so its last digit must be even.
The last digit here is A, so A must be even.
Of the choices 3, 6, 9, only 6 is even.
Therefore $A = 6$.

$$A \text{ even and } A \in \{3,6,9\} \;\Rightarrow\; A = 6$$

💡 Divisible by 4 forces an even last digit, and 6 is the only even option left for A.

#4 Introduce a Variable 6.EE.B.6 Step 5

Narrow B with the rule for 9

Put $A = 6$ back into the digit sum: $2(6) + 3B = 12 + 3B$ must be a multiple of 9.
Dividing the condition by 3, we need $4 + B$ to be a multiple of 3, so B leaves remainder 2 when divided by 3.
The digit choices are $B \in \{2, 5, 8\}$.

$$12 + 3B \equiv 0 \pmod 9 \;\Rightarrow\; B \equiv 2 \pmod 3 \;\Rightarrow\; B \in \{2,5,8\}$$

💡 Locking A at 6 turns the rule for 9 into a short list of possible B values.

#3 Eliminate Possibilities 4.OA.B.4 Step 6

Finish with the rule for 4

The rule for 4 checks the last two digits, which are B then A, forming the number $10B + A = 10B + 6$.
Test each candidate: $B = 2$ gives 26 (not divisible by 4), $B = 5$ gives 56 (yes, $56 = 4 \times 14$), $B = 8$ gives 86 (not divisible by 4).
Only $B = 5$ works.
So $A = 6$ and $B = 5$, giving $A + B = 11$.
The answer is (C).

$$10B + 6 \equiv 0 \pmod 4 \;\Rightarrow\; B = 5;\quad A + B = 6 + 5 = 11$$

💡 The last two digits '56' are the only ones divisible by 4, which uniquely fixes B.

[1] #8 5.NBT.A.3 The total is a money amount with the digits A, B, B before the point and B, A af

[2] #3 4.OA.B.4 A number is divisible by 36 exactly when it is divisible by both 4 and 9, since

[3] #4 6.EE.B.6 The rule for 9 says the digit sum must be a multiple of 9. The digits are A, B,

[4] #3 4.OA.B.4 A number divisible by 4 is even, so its last digit must be even. The last digit

[5] #4 6.EE.B.6 Put $A = 6$ back into the digit sum: $2(6) + 3B = 12 + 3B$ must be a multiple of

[6] #3 4.OA.B.4 The rule for 4 checks the last two digits, which are B then A, forming the numbe

Review

Reasonableness: Check the found number directly: with $A = 6$, $B = 5$ the cents total is 65,556. Divide by 36: $65556 \div 36 = 1821$, a whole number, so each shirt costs $18.21 and the split is exact. The digit sum $6+5+5+5+6 = 27$ is a multiple of 9, and the last two digits 56 are a multiple of 4, confirming divisibility by 36. So $A + B = 11$, matching choice (C).

Alternative: Instead of reasoning with the digit rules, list the finitely many candidates by brute force: A and B are each single digits with $A \neq 0$, so loop over the numbers $A\,B\,B\,B\,A$ and keep only those divisible by 36. The single surviving number is 65,556, again giving $A + B = 11$.

CCSS standards used (min grade 6)

5.NBT.A.3 Read, write, and compare decimals to thousandths (Reading the dollars-and-cents amount $A\,B\,B.\,B\,A$ and converting it into a whole number of cents so divisibility makes sense.)
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite (Recognizing $36 = 4 \times 9$ and applying the divisibility rules for 4 and 9 to eliminate digit candidates.)
6.EE.B.6 Use variables to represent numbers and write expressions to solve problems (Letting the digits A and B stand for unknowns and building the expression $2A + 3B$ for the digit sum.)

⭐ To check divisibility by 36, split it into 4 and 9 and use the two easy digit rules to cross off every impossible pair until one survives.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Read the price as whole cents

Split 36 into 4 and 9

Apply the rule for 9

Use divisibility by 4 to pin A

Narrow B with the rule for 9

Finish with the rule for 4

Review

CCSS standards used (min grade 6)

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