AMC 10 · 2021 · #3

Grade 6 number-theory

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

place-valuedigit-decompositionlinear-equations-one-varmulti-digit-arithmetic convert-to-algebraidentify-subproblems ↑ Prerequisites: place-value

📏 Short solution 💡 2 insights

Problem

Two natural numbers add to 17,402. The bigger one ends in 0 (it is a multiple of 10). Erasing that bigger number's units digit leaves the smaller number. Find the difference between the two numbers.

Pick an answer.

(A)

$~10{,}272$

(B)

$~11{,}700$

(C)

$~13{,}362$

(D)

$~14{,}238$

(E)

$~15{,}426$

AMC 10 2021 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

Tool #7 (Subproblems) is the spine: name the smaller number s; then the bigger is 10s (since erasing the ones digit is the same as dividing by 10). The sum condition becomes one tiny equation 11s = 17,402, and the difference is 9s. Tool #3 (Eliminate) acts as a fast safety net — the difference ends in 9s, whose ones digit must be 8 (since s ends in 2). Only choice (D) 14,238 ends in 8.

1STEP 1

Call the smaller number s. Erasing the multiple-of-10's units digit divides by 10, so the bigger number is 10s.

smaller = s, bigger = 10s

2STEP 2

Sum condition: 10s + s = 11s = 17,402, so s = 17,402 ÷ 11 = 1,582.

11s = 17,402 → s =

\frac{17,402}{11}

= 1,582

3STEP 3

Then bigger = 10 × 1,582 = 15,820, and the difference 15,820 - 1,582 = 14,238 — choice (D).

15,820 - 1,582 = 14,238 → (D)

4STEP 4

Safety check: smaller ends in 2, bigger in 0, so the difference's ones digit is 0 - 2 = 8 — only (D) ends in 8.

ones digit of difference = 8 → (D)

Answer

~14,238

Verify the trio: smaller + bigger = 1,582 + 15,820 = 17,402 ✓. Erasing the ones digit of 15,820 gives 1,582 ✓. Difference 14,238 is roughly

\frac{9}{11}

of 17,402 ≈ 14,238 ✓. The number lines up with choice (D), and no other choice has the right ones digit.

💡Key takeaway

This AMC 10 problem only needs Grade 6 "sliding digits left is multiplying by ten" — once you call the smaller number s, the bigger is 10s, and 11s = 17,402 gives s = 1,582, so the difference is 14,238.

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