AMC 10 · 2024 · #8

Grade 8 arithmetic

Archetype Units-Digit Tracking └ Sub-archetype Power Cycle Lookup

divisor-countfactorsunits-digit-trackingprime-factorization identify-subproblemsunits-digit-trackingmodular-arithmetic-mod-10 ↑ Prerequisites: factorsprime-factorizationunits-digit-tracking

📏 Short solution 💡 2 insights

Problem

Multiply every positive integer that divides 42 together. Call this product N. Find the units digit (ones digit) of N.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Look for a Pattern

The pattern Tool #5 spots is the symmetry of divisors: every divisor d has a unique "partner" $\frac{42}{d}$ , and the two multiply to 42. So once we list the divisors systematically (Tool #2), we can pair them into 42 × 42 × 42 × 42 = 42⁴. Tool #7 (Identify Subproblems) then reduces the units-digit question to just "what is the ones digit of 42⁴?" — which depends only on the ones digit 2 raised to the 4th power.

1STEP 1

Since 42 = 2 × 3 × 7, every divisor is a subset-product of {2, 3, 7}, giving 8 divisors in all.

divisors of 42: 1, 2, 3, 6, 7, 14, 21, 42

2STEP 2

Match each small divisor with the partner that multiplies to 42; the eight divisors split into 4 such pairs.

1 · 42 = 42, 2 · 21 = 42, 3 · 14 = 42, 6 · 7 = 42

3STEP 3

Each of the 4 pairs contributes a factor of 42, so the whole product collapses to 42⁴.

N = 1 · 2 · 3 · 6 · 7 · 14 · 21 · 42 = (1 · 42)(2 · 21)(3 · 14)(6 · 7) = 42⁴

4STEP 4

A power's ones digit tracks the base's ones digit, so 42⁴ ends in the ones digit of 2⁴ = 16, namely 6 — choice (D).

42⁴ ≡ 2⁴ ≡ 16 ≡ 6 (mod 10) → (D)

Answer

Cross-check by computing the units digit the long way. Multiply the ones digits in order: 1 · 2 · 3 · 6 · 7 · 4 · 1 · 2. Step by step (keeping only ones digits): 1 → 2 → 6 → 6 → 2 → 8 → 8 → 6. Ones digit = 6, matching answer (D). The pairing trick gives the same answer with far less arithmetic.

💡Key takeaway

This AMC 10 problem only needs Grade 8 exponent rules — pair the divisors of 42 so each pair multiplies to 42, and the ones digit of 42⁴ is the ones digit of 2⁴ = 16!

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