Competition · AMC preparation · step 4 of 4

AMC 8 · 2013 · #13

Grade 5 number-theoryarithmetic
Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search
digit-decompositionplace-valuemultiples convert-to-algebrasystematic-enumeration ↑ Prerequisites: place-valuemulti-digit-arithmetic
📏 Medium solution 💡 3 insights
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Problem
Clara added up several test scores but, on one score, she accidentally swapped the tens digit and the units digit (for example, she wrote 72 as 27). Because of that single swap, her total came out different from the true total. Which of the five answer choices could that difference be?

Pick an answer.

(A)
45
(B)
46
(C)
47
(D)
48
(E)
49

AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Solve an Easier Related Problem

Instead of jumping into algebra, try small concrete cases first (Tool #9). Reverse the digits of a few two-digit numbers — 72 → 27, 63 → 36, 41 → 14 — and write down each difference. Tool #5 (Look for a Pattern) spots that every difference is a multiple of 9. Tool #3 (Eliminate Possibilities) then sweeps the five answer choices and keeps only the multiple of 9. This path uses Grade 4 multiples / place-value reasoning instead of Grade 6 algebra.

1STEP 1

Try a few examples

Work a few cases by hand: 72 - 27 = 45, 63 - 36 = 27, 41 - 14 = 27.

72 - 27 = 45, 63 - 36 = 27, 41 - 14 = 27
2STEP 2

Look for the common factor

Every difference — 45, 27, 27 — is a multiple of 9; one more case, 85 - 58 = 27, confirms it.

45 = 9 × 5, 27 = 9 × 3, 85 - 58 = 27 = 9 × 3
3STEP 3

Explain it with place value

Place value proves it: (10t + u) - (10u + t) = 9(t - u), always a multiple of 9.

(10t + u) - (10u + t) = 9t - 9u = 9(t - u)
4STEP 4

Keep only multiples of 9

Test the five choices against the 9-rule; only 45 is a multiple of 9 (4+5=9).

45 = 9 × 5 ✓; 46, 47, 48, 49 not divisible by 9
5STEP 5

Confirm 45 really happens

Check it's reachable: 9|t - u| = 45 needs |t - u| = 5, e.g. 72 - 27 = 45.

9 × |t - u| = 45 → |t - u| = 5; 72 - 27 = 45 → (A)
Answer
45
Every test case we tried — 72 → 27, 63 → 36, 41 → 14, 85 → 58 — gave a difference that is a multiple of 9. The algebraic shortcut 9(t-u) explains why this must be true. Among the five choices, only 45 is a multiple of 9, and we exhibited a real swap (72 ⇔ 27) that produces exactly 45. The reasoning and the example both point to (A).
💡Key takeaway

Swapping the two digits of a number always changes it by a multiple of 9 — a Grade 5 place-value fact, not an AMC mystery!

  • Try a few examples
  • Look for the common factor
  • Explain it with place value
  • Keep only multiples of 9
  • Confirm 45 really happens

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