AMC 8 · 2016 · #9

Grade 5 number-theory

Archetype Arithmetic Expression Decomposition

prime-factorizationdivisibility-rulesprime-numbers identify-subproblemsguess-and-check ↑ Prerequisites: divisibility-rulesprime-numbers

📏 Short solution 💡 2 insights

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Problem

What is the sum of the distinct prime integer divisors of $2016$ ?

$\textbf{(A) }9\qquad\textbf{(B) }12\qquad\textbf{(C) }16\qquad\textbf{(D) }49\qquad \textbf{(E) }63$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: Find the sum of the distinct prime numbers that divide $2016$. "Distinct" means each prime is counted once, no matter how many times it appears in the prime factorization.

Givens: The target number is $2016$; We want only the prime divisors (not all divisors); We want distinct primes — repeats are counted once; Answer choices: (A) $9$, (B) $12$, (C) $16$, (D) $49$, (E) $63$

Unknowns: The sum of the distinct prime divisors of $2016$

Understand

Restated: Find the sum of the distinct prime numbers that divide $2016$. "Distinct" means each prime is counted once, no matter how many times it appears in the prime factorization.

Plan

Primary tool: #7 Identify Subproblems

Secondary: #6 Guess and Check

Tool #7 (Identify Subproblems) splits the question into two clean jobs: (a) factor $2016$ into primes, then (b) add the distinct prime bases. Tool #6 (Guess and Check) handles step (a) — trial-divide by the small primes $2, 3, 5, 7, \ldots$ in order until the quotient becomes $1$. We deliberately avoid heavier tools like #13 (Algebra) because plain trial division is the most direct path for a four-digit number.

Execute — Answer: B

#6 Guess and Check 5.NBT.B.6 Step 1

Subproblem 1: factor $2016$.
Start by dividing out the smallest prime, $2$, as many times as possible.
$2016$ is even, so we keep halving until we hit an odd number.

$$2016 \div 2 = 1008,\;\; 1008 \div 2 = 504,\;\; 504 \div 2 = 252,\;\; 252 \div 2 = 126,\;\; 126 \div 2 = 63$$

💡 Repeated division by $2$ is a Grade 5 multi-digit division skill — no special technique needed.

#6 Guess and Check 4.OA.B.4 Step 2

$63$ is odd, so $2$ is finished.
Try the next prime, $3$.
The digit sum of $63$ is $6+3=9$, which is a multiple of $3$, so $3$ divides $63$.
Keep dividing by $3$ until you cannot anymore.

$$63 \div 3 = 21,\;\; 21 \div 3 = 7$$

💡 Using the digit-sum rule for $3$ and recognizing prime factors is exactly the Grade 4 "factors and multiples" standard.

#7 Identify Subproblems 4.OA.B.4 Step 3

The remaining quotient $7$ is itself prime, so the factorization is complete.
Write out the full prime factorization, collecting like primes into exponents.

$$2016 = 2^5 \times 3^2 \times 7$$

💡 Recognizing that $7$ is prime ends the factorization — a Grade 4 prime/composite check.

#7 Identify Subproblems 4.OA.B.4 Step 4

Subproblem 2: list the distinct prime bases (ignore the exponents — they only tell us how many times each prime appears, not which primes appear).
The distinct primes are $2$, $3$, and $7$.

$$\text{Distinct primes} = \{2,\, 3,\, 7\}$$

💡 "Distinct" means we list each prime base once — a careful reading of the problem, not a calculation.

#7 Identify Subproblems 2.NBT.B.5 Step 5

Add the three distinct primes and match the result to the answer choices.

$$2 + 3 + 7 = 12 \;\Rightarrow\; \textbf{(B)}$$

💡 Adding three small whole numbers is a Grade 2 fluency skill.

[1] #6 5.NBT.B.6 Subproblem 1: factor $2016$. Start by dividing out the smallest prime, $2$, as m

[2] #6 4.OA.B.4 $63$ is odd, so $2$ is finished. Try the next prime, $3$. The digit sum of $63$

[3] #7 4.OA.B.4 The remaining quotient $7$ is itself prime, so the factorization is complete. Wr

[4] #7 4.OA.B.4 Subproblem 2: list the distinct prime bases (ignore the exponents — they only te

[5] #7 2.NBT.B.5 Add the three distinct primes and match the result to the answer choices.

Review

Reasonableness: Multiply the factorization back together to confirm: $2^5 = 32$, $3^2 = 9$, and $32 \times 9 = 288$, then $288 \times 7 = 2016$. The factorization is correct, so the distinct primes are truly $\{2, 3, 7\}$ and their sum is $12$. Choice (B) is also a small, sensible number — the trap answers (D) $49$ and (E) $63$ correspond to adding the prime powers ($2^5 + 3^2 + 7 = 32 + 9 + 7 = 48$, close to $49$) or to the last factor before $7$ ($63$), both common misreadings.

Alternative: Tool #9 (Solve an Easier Related Problem): break $2016$ into familiar pieces — $2016 = 2 \times 1008 = 2 \times 2 \times 504 = \ldots$ or notice $2016 = 2000 + 16 = 16 \times 126 = 16 \times 2 \times 63 = 32 \times 63$. Then $32 = 2^5$ and $63 = 9 \times 7 = 3^2 \times 7$, so $2016 = 2^5 \times 3^2 \times 7$. Same distinct primes, same sum $2+3+7 = 12$.

CCSS standards used (min grade 5)

2.NBT.B.5 Fluently add and subtract within 100 (Adding the three distinct primes $2 + 3 + 7 = 12$ to produce the final answer.)
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite (Recognizing $2$, $3$, and $7$ as prime divisors and using the digit-sum rule to test divisibility by $3$.)
5.NBT.B.6 Find whole-number quotients of whole numbers with up to four-digit dividends (Performing the repeated divisions $2016 \div 2 = 1008$, $1008 \div 2 = 504$, $\ldots$ to peel off the prime factors.)

⭐ This AMC 8 problem only needs Grade 5 division and the Grade 4 idea of prime factors that you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 5)

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