AMC 8 · 2013 · #15
Easy mode Grade 6Problem
Three different equations are given, each with a missing exponent.
- The first equation: 3p+34=90. Find the whole number p that makes this true.
- The second equation: 2r+44=76. Find the whole number r that makes this true.
- The third equation: 53+6s=1421. Find the whole number s that makes this true.
Once you have p, r, and s, multiply them together.
What is the value of p×r×s?
Pick an answer.
AMC 8 2013 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: Three separate equations each hide one exponent: $3^p + 3^4 = 90$, $2^r + 44 = 76$, and $5^3 + 6^s = 1421$. Find $p$, $r$, and $s$, then compute the product $p \cdot r \cdot s$.
Givens: $3^p + 3^4 = 90$; $2^r + 44 = 76$; $5^3 + 6^s = 1421$; Answer choices: (A) $27$, (B) $40$, (C) $50$, (D) $70$, (E) $90$
Unknowns: The exponents $p$, $r$, $s$; Their product $p \cdot r \cdot s$
Understand
Restated: Three separate equations each hide one exponent: $3^p + 3^4 = 90$, $2^r + 44 = 76$, and $5^3 + 6^s = 1421$. Find $p$, $r$, and $s$, then compute the product $p \cdot r \cdot s$.
Givens: $3^p + 3^4 = 90$; $2^r + 44 = 76$; $5^3 + 6^s = 1421$; Answer choices: (A) $27$, (B) $40$, (C) $50$, (D) $70$, (E) $90$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #6 Guess and Check
The three equations have no shared variables, so Tool #7 (Identify Subproblems) is the obvious move — solve each equation by itself, then multiply at the end. Inside each subproblem, isolate the power of the base on one side and ask "which exponent of this base gives that number?" Listing $b^1, b^2, b^3, \ldots$ until you hit the target is Tool #6 (Guess and Check) on a tiny ladder of values.
Execute — Answer: B
6.EE.A.1 Step 1 - Solve the first equation for $p$.
- Evaluate $3^4 = 81$ and subtract from both sides, leaving a clean power of $3$ on the left.
💡 Recognizing $9$ as $3^2$ is the Grade 6 "whole-number exponents" skill.
6.EE.A.1 Step 2 - Solve the second equation for $r$.
- Subtract $44$ from both sides, then match the result to a power of $2$.
💡 Walking up the powers of $2$ ($2, 4, 8, 16, 32$) until you land on $32$ is Tool #6 in miniature.
6.EE.A.1 Step 3 - Solve the third equation for $s$.
- Evaluate $5^3 = 125$, subtract from both sides, then test powers of $6$ until one matches.
💡 $1296$ isn't an instantly recognizable number, but four short multiplications by $6$ pin it down — exactly the Tool #6 pattern.
3.OA.A.1 Step 4 Multiply the three exponents to get the requested product.
💡 After the subproblems are finished, the closing move is just a Grade 3 multiplication of whole numbers.
6.EE.A.1 Solve the first equation for $p$. Evaluate $3^4 = 81$ and subtract from both sid 6.EE.A.1 Solve the second equation for $r$. Subtract $44$ from both sides, then match the 6.EE.A.1 Solve the third equation for $s$. Evaluate $5^3 = 125$, subtract from both sides 3.OA.A.1 Multiply the three exponents to get the requested product. Review
Reasonableness: Substitute back: $3^2 + 3^4 = 9 + 81 = 90$ ✓, $2^5 + 44 = 32 + 44 = 76$ ✓, $5^3 + 6^4 = 125 + 1296 = 1421$ ✓. The product $2 \cdot 5 \cdot 4 = 40$ matches choice (B), and it's the only answer choice that factors into three small positive integers consistent with these equations.
Alternative: Tool #6 (Guess and Check) directly against the answer choices. The three exponents must be positive integers whose product is one of $27, 40, 50, 70, 90$. From the second equation alone, $2^r = 32$ forces $r = 5$, so the product must be a multiple of $5$ — eliminating (A) $27$ and (D) $70 / 5 = 14$ (not factorable as $p \cdot s$ with $p \leq 4$, $s \leq 4$). Then $p = 2, s = 4$ gives $2 \cdot 5 \cdot 4 = 40$, confirming (B).
CCSS standards used (min grade 6)
3.OA.A.1Interpret products of whole numbers (Computing the final product $2 \times 5 \times 4 = 40$ once the three exponents are known.)6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents (Evaluating $3^4 = 81$, $5^3 = 125$, and matching $9 = 3^2$, $32 = 2^5$, $1296 = 6^4$ to recover each exponent.)
⭐ Three equations, one trick: peel off the constant, then ask "which power of this base gives that number?" — all Grade 6 exponent work.
⭐ Three equations, one trick: peel off the constant, then ask "which power of this base gives that number?" — all Grade 6 exponent work.
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