AMC 8 · 2006 · #24

Grade 6 number-theory

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

digit-constraintsdigit-decompositionplace-valuelogical-deduction pattern-recognitiondigit-constraints ↑ Prerequisites: place-valuemulti-digit-arithmetic

📏 Short solution 💡 2 insights

Problem

In the multiplication ABA × CD = CDCD, the letters A, B, C, D are four different digits. Find A + B.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Look for a Pattern

The number CDCD is just the two-digit block CD written twice. Tool #5 (Look for a Pattern) spots that this repetition is the same trick as abab = ab × 101 (similar to how aa = a × 11). Once we name that pattern, Tool #13 (Convert to Algebra) turns the multiplication line into a one-line equation ABA × CD = CD × 101, which we can divide by CD to read ABA directly. No guessing needed.

1STEP 1

Spot that CDCD is the block CD copied twice, so place value expands it to CD × 101.

CDCD = (10C + D) · 100 + (10C + D) = (10C + D) · 101 = CD · 101

2STEP 2

Turn the multiplication into an equation and substitute Step 1: ABA × CD = CD × 101.

ABA × CD = CDCD = CD · 101

3STEP 3

Divide both sides by CD, which is nonzero since C ≠ 0, giving ABA = 101.

ABA =

\frac{CD · 101}{CD}

= 101

4STEP 4

Read the digits of 101 as A = 1 and B = 0, so A + B = 1.

A + B = 1 + 0 = 1 ⟹ (A)

Answer

Test the answer with a concrete choice of C and D. Pick C = 2, D = 3, so CD = 23. Then ABA × 23 = 101 × 23 = 2323 = CDCD, exactly as required. Digits used: A = 1, B = 0, C = 2, D = 3 — all four different. The product checks out, and A + B = 1 matches choice (A). Try C = 5, D = 7 to be sure: 101 × 57 = 5757 — same pattern, same A + B. The answer is independent of the specific C, D chosen, which is exactly why the problem can be solved without finding them.

💡Key takeaway

When a block of digits repeats — like CDCD being CD written twice — it always factors out as that block times 101. Spot that pattern and the multiplication puzzle collapses to a Grade 6 one-step equation.

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