AMC 8 · 2007 · #10

Grade 5 number-theory

Archetype Arithmetic Expression Decomposition

factorsdivisor-countmulti-digit-arithmetic identify-subproblems ↑ Prerequisites: factorsmulti-digit-arithmetic

📏 Short solution 💡 2 insights

Problem

For any positive integer $n$ , define $\boxed{n}$ to be the sum of the positive factors of $n$ .
For example, $\boxed{6} = 1 + 2 + 3 + 6 = 12$ . Find $\boxed{\boxed{11}}$ .

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: Define $\boxed{n}$ to be the sum of all positive factors of $n$. For example, $\boxed{6} = 1 + 2 + 3 + 6 = 12$. Compute $\boxed{\boxed{11}}$.

Givens: $\boxed{n}$ means the sum of the positive factors of $n$; Example: $\boxed{6} = 1 + 2 + 3 + 6 = 12$; We must evaluate $\boxed{\boxed{11}}$ — the same operation applied twice, starting at $11$; Answer choices: (A) $13$, (B) $20$, (C) $24$, (D) $28$, (E) $30$

Unknowns: The value of $\boxed{\boxed{11}}$

Understand

Restated: Define $\boxed{n}$ to be the sum of all positive factors of $n$. For example, $\boxed{6} = 1 + 2 + 3 + 6 = 12$. Compute $\boxed{\boxed{11}}$.

Plan

Primary tool: #7 Identify Subproblems

Secondary: #2 Make a Systematic List

Tool #7 (Identify Subproblems) splits the nested expression $\boxed{\boxed{11}}$ into two single-step questions: first $\boxed{11}$ (which becomes a number), then $\boxed{\text{that number}}$. Each piece is just "list the factors and add them." Tool #2 (Make a Systematic List) is how we collect each factor exactly once — try divisors $1, 2, 3, \ldots$ up to $n$ in order, keep the ones that divide evenly. No algebra, no formulas; the two subproblems are both Grade 4 work.

Execute — Answer: D

#2 Make a Systematic List 4.OA.B.4 Step 1

Evaluate the inner box: $\boxed{11}$.
List every positive factor of $11$.
Since $11$ is prime, only $1$ and $11$ divide it evenly.

$$\boxed{11} = 1 + 11 = 12$$

💡 A prime number has exactly two factors: $1$ and itself. That makes the inner box the easiest possible case.

#7 Identify Subproblems 5.OA.A.1 Step 2

Replace the inner box with its value.
The expression $\boxed{\boxed{11}}$ now becomes $\boxed{12}$.

$$\boxed{\boxed{11}} = \boxed{12}$$

💡 Working from the inside of a nested expression outward is the standard "parentheses first" order-of-operations move.

#2 Make a Systematic List 4.OA.B.4 Step 3

Evaluate the outer box: $\boxed{12}$.
Walk the divisors $1, 2, 3, 4, 5, 6, \ldots$ in order and keep the ones that divide $12$ evenly.
The factors of $12$ are $1, 2, 3, 4, 6, 12$ (note $5$ does not divide $12$; after $6$ the next factor would be $12 \div 1 = 12$).

$$\boxed{12} = 1 + 2 + 3 + 4 + 6 + 12 = 28$$

💡 Listing factors in order from $1$ upward guarantees none are missed. Stopping at $\sqrt{12} \approx 3.5$ and pairing each small factor with $12 \div \text{factor}$ also gives the same six numbers.

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

The outer sum is the answer.

$$\boxed{\boxed{11}} = 28 \;\Rightarrow\; \textbf{(D)}$$

💡 Adding the six factors of $12$ in order: $1+2 = 3$, $3+3 = 6$, $6+4 = 10$, $10+6 = 16$, $16+12 = 28$. Choice (D).

[1] #2 4.OA.B.4 Evaluate the inner box: $\boxed{11}$. List every positive factor of $11$. Since

[2] #7 5.OA.A.1 Replace the inner box with its value. The expression $\boxed{\boxed{11}}$ now be

[3] #2 4.OA.B.4 Evaluate the outer box: $\boxed{12}$. Walk the divisors $1, 2, 3, 4, 5, 6, \ldot

[4] #7 4.NBT.B.4 The outer sum is the answer.

Review

Reasonableness: Re-pair the factors of $12$ to double-check the sum: $1 \cdot 12 = 12$, $2 \cdot 6 = 12$, $3 \cdot 4 = 12$. Three pairs that each multiply to $12$, with sums $1+12 = 13$, $2+6 = 8$, $3+4 = 7$. Total: $13 + 8 + 7 = 28$. The answer (D) matches, and the size feels right — $\boxed{12}$ should be a bit more than $12$ itself (since the proper factors $1, 2, 3, 4, 6$ alone already add to $16$). Also, $5$ correctly drops out because $12 / 5$ is not a whole number.

Alternative: Tool #3 (Eliminate Possibilities): once $\boxed{11} = 12$ is in hand, the question reduces to "sum of factors of $12$." The factors must include $1$ and $12$ themselves, so the total is at least $13$. That already rules out (A) $13$ (which would force $12$ to be prime — false, since $2$ divides $12$). Adding the obvious divisors $2$ and $3$ pushes the total to at least $1 + 2 + 3 + 12 = 18$. Adding $4$ and $6$ brings it to $28$. (D) is the only choice consistent with all six factors.

CCSS standards used (min grade 5)

4.OA.B.4 Find factor pairs, recognize multiples, and identify prime numbers within $1$-$100$ (Listing the factors of $11$ (just $1$ and $11$, since $11$ is prime) and the factors of $12$ ($1, 2, 3, 4, 6, 12$) — the Grade 4 factor-pairs and primes standard.)
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols (Evaluating the nested expression $\boxed{\boxed{11}}$ from the inside out — the Grade 5 grouping-symbols standard governs this innermost-first order.)
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm (Summing the six factors of $12$: $1 + 2 + 3 + 4 + 6 + 12 = 28$.)

⭐ When a symbol is wrapped inside itself, evaluate the inside first and substitute — then this AMC 8 problem is just two Grade 4 factor lists in a row.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 5)

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