AMC 10 · 2025 · #6

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

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.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Eliminate Possibilities

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.

1STEP 1

Read the price as whole cents

Drop the point and read whole cents.

cents = underline A underline B underline B underline B underline A, 36 ∣ cents

2STEP 2

Split 36 into 4 and 9

36 is four times nine.

36 = 4 × 9

3STEP 3

Apply the rule for 9

The nine rule makes A a multiple of three.

2A + 3B ≡ 0 (mod 9) → 3 ∣ A → A ∈ {3,6,9}

4STEP 4

Use divisibility by 4 to pin A

The four rule makes it even, so A is 6.

A even and A ∈ {3,6,9} → A = 6

5STEP 5

Narrow B with the rule for 9

The nine rule leaves B as 2, 5, or 8.

12 + 3B ≡ 0 (mod 9) → B ≡ 2 (mod 3) → B ∈ {2,5,8}

6STEP 6

Finish with the rule for 4

The four rule fixes B at five, so the sum is 11.

10B + 6 ≡ 0 (mod 4) → B = 5; A + B = 6 + 5 = 11

Answer

Check the found number directly: with A = 6, B = 5 the cents total is 65,556. Divide by 36: 65556 ÷ 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).

💡Key takeaway

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.

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