AMC 10 · 2024 · #8
Grade 8 arithmeticProblem
Let N be the product of all the positive integer divisors of 42. What is the units digit
of N?
Pick an answer.
AMC 10 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Multiply every positive integer that divides $42$ together. Call this product $N$. Find the units digit (ones digit) of $N$.
Givens: $N = $ product of all positive divisors of $42$; $42 = 2 \times 3 \times 7$, so the divisors are $1, 2, 3, 6, 7, 14, 21, 42$; Answer choices: (A) $0$, (B) $2$, (C) $4$, (D) $6$, (E) $8$
Unknowns: The ones digit of the (very large) number $N$
Understand
Restated: Multiply every positive integer that divides $42$ together. Call this product $N$. Find the units digit (ones digit) of $N$.
Givens: $N = $ product of all positive divisors of $42$; $42 = 2 \times 3 \times 7$, so the divisors are $1, 2, 3, 6, 7, 14, 21, 42$; Answer choices: (A) $0$, (B) $2$, (C) $4$, (D) $6$, (E) $8$
Plan
Primary tool: #5 Look for a Pattern
Secondary: #2 Make a Systematic List, #7 Identify Subproblems
The pattern Tool #5 spots is the symmetry of divisors: every divisor $d$ has a unique "partner" $42/d$, and the two multiply to $42$. So once we list the divisors systematically (Tool #2), we can pair them into $42 \times 42 \times 42 \times 42 = 42^4$. Tool #7 (Identify Subproblems) then reduces the units-digit question to just "what is the ones digit of $42^4$?" — which depends only on the ones digit $2$ raised to the $4$th power.
Execute — Answer: D
4.OA.B.4 Step 1 - List every positive divisor of $42$ in order.
- Since $42 = 2 \times 3 \times 7$, every divisor is some subset-product of $\{2, 3, 7\}$, giving $2 \times 2 \times 2 = 8$ divisors total.
💡 Listing every divisor in order is exactly the Grade 4 "find all factor pairs" move — every divisor has to show up here.
4.OA.B.4 Step 2 - Pair the divisors.
- Match each small divisor with its large partner whose product is $42$.
- The $8$ divisors break into $4$ partner pairs.
💡 Divisors come in pairs whose product is the original number — Tool #5 spots this symmetry and Grade 4 "factor pair" thinking confirms it.
6.EE.A.1 Step 3 - Multiply all four pair-products together.
- Since each of the $4$ pairs contributes a factor of $42$, the full product $N$ is $42$ raised to the $4$th power.
💡 Four pairs each equaling $42$ multiply to $42^4$ — Grade 6 "evaluate expressions with whole-number exponents".
8.EE.A.1 Step 4 - Now find only the units digit of $42^4$.
- The ones digit of a power depends only on the ones digit of the base, so the ones digit of $42^4$ equals the ones digit of $2^4$.
💡 Tens-and-higher digits cannot affect the ones digit of a product — Grade 8 integer-exponent properties make this routine.
4.OA.B.4 List every positive divisor of $42$ in order. Since $42 = 2 \times 3 \times 7$, 4.OA.B.4 Pair the divisors. Match each small divisor with its large partner whose product 6.EE.A.1 Multiply all four pair-products together. Since each of the $4$ pairs contribute 8.EE.A.1 Now find only the units digit of $42^4$. The ones digit of a power depends only Review
Reasonableness: Cross-check by computing the units digit the long way. Multiply the ones digits in order: $1 \cdot 2 \cdot 3 \cdot 6 \cdot 7 \cdot 4 \cdot 1 \cdot 2$. Step by step (keeping only ones digits): $1 \to 2 \to 6 \to 6 \to 2 \to 8 \to 8 \to 6$. Ones digit $= 6$, matching answer (D). The pairing trick gives the same answer with far less arithmetic.
Alternative: Tool #2 (Systematic List) alone, skipping the pairing trick: multiply the ones digits left to right as just shown, tracking only the current ones digit. The pairing trick is faster, but the brute list-and-multiply path also lands on $6$, confirming answer (D).
CCSS standards used (min grade 8)
4.OA.B.4Find all factor pairs and recognize multiples; determine prime or composite (Listing all $8$ divisors of $42$ and pairing each small divisor $d$ with its partner $42/d$ so every pair multiplies to $42$.)6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents (Combining the four pair-products $42 \cdot 42 \cdot 42 \cdot 42$ into the single expression $42^4$.)8.EE.A.1Know and apply the properties of integer exponents (Using $42^4 \equiv 2^4 = 16 \pmod{10}$ — the ones digit of a power depends only on the ones digit of the base — to extract the units digit $6$.)
⭐ 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 $42^4$ is the ones digit of $2^4 = 16$!
⭐ 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 $42^4$ is the ones digit of $2^4 = 16$!
More like this
Same archetype — closest grade level first.