AMC 10 · 2011 · #5

Grade 4 number-theory

Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration

prime-factorizationdigit-decompositionplace-value work-backwards ↑ Prerequisites: prime-factorization

📏 Medium solution 💡 2 insights

Problem

Ron multiplied a two-digit number a by another positive integer b, but he flipped the two digits of a before multiplying. That wrong multiplication gave 161. Using the correct value of a, find the true product a × b.

Pick an answer.

(A)

116

(B)

161

(C)

204

(D)

214

(E)

224

AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Work Backwards

The wrong product 161 is the end of Ron's mistake, and we need the numbers he started from, so Tool #11 (Work Backwards) fits: factor 161 to recover the two numbers he actually multiplied. Tool #3 (Eliminate Possibilities) then picks which factor is the reversed two-digit number — only one factor has two digits. Tool #7 (Identify Subproblems) breaks the rest into small steps: un-flip the digits to get a, then do the correct multiplication.

1STEP 1

Factor the wrong product

The wrong product (reversed a) × b = 161 is a whole-number factor pair, so factor it: 161 = 7 × 23, both prime.

161 = 7 × 23

2STEP 2

Pick the reversed two-digit factor

A flipped two-digit number still has two digits, so reversed a = 23 and b = 7; the split 1 × 161 is impossible.

reversed a = 23, b = 7

3STEP 3

Un-flip the digits to get a

Flip 23 back — the tens and ones digits swap — recovering the true a = 32, with b = 7.

23 → a = 32

4STEP 4

Multiply correctly

Now multiply the real numbers: a × b = 32 × 7 = 210 + 14 = 224, which is choice (E), not the bait 161.

a × b = 32 × 7 = 224 → (E)

Answer

224

Check the story both ways. Reversed a = 23 times b = 7 gives 23 × 7 = 161, exactly Ron's wrong product, so the factoring is right. The true a = 32 times b = 7 gives 224, which is close to but a bit larger than 161 — sensible, since 32 ＞ 23. The answer 224 is choice (E), and the tempting 161 (choice B) is only the mistaken product, not the correct one.

💡Key takeaway

Break the wrong answer into its factors, flip the two-digit one back to the real number, then multiply again to get the true product.

Factor the wrong product
Pick the reversed two-digit factor
Un-flip the digits to get a
Multiply correctly

Factor the wrong product

Pick the reversed two-digit factor

Un-flip the digits to get a

Multiply correctly

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