AMC 8 · 2018 · #25

Grade 8 number-theory

Archetype Small-Domain Constrained Search

exponentsperfect-squaresbound-inequality-then-enumerate bound-inequality-then-enumeratesystematic-enumeration ↑ Prerequisites: exponentsmulti-digit-arithmetic

📏 Short solution 💡 2 insights

Problem

How many perfect cubes lie between $2^8+1$ and $2^{18}+1$ , inclusive?

$\textbf{(A) }4\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }57\qquad \textbf{(E) }58$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: Count every perfect cube $n^3$ (where $n$ is a positive integer) that lies in the closed interval $[2^8+1,\; 2^{18}+1]$. The endpoints themselves count if they happen to be perfect cubes.

Givens: Lower bound: $2^8 + 1 = 257$; Upper bound: $2^{18} + 1$; Interval is inclusive on both ends; Answer choices: (A) $4$, (B) $9$, (C) $10$, (D) $57$, (E) $58$

Unknowns: The number of integers $n \ge 1$ such that $2^8 + 1 \le n^3 \le 2^{18} + 1$

Understand

Givens: Lower bound: $2^8 + 1 = 257$; Upper bound: $2^{18} + 1$; Interval is inclusive on both ends; Answer choices: (A) $4$, (B) $9$, (C) $10$, (D) $57$, (E) $58$

Plan

Primary tool: #7 Identify Subproblems

Secondary: #6 Guess and Check, #9 Solve an Easier Related Problem

"How many cubes lie in $[L, U]$?" naturally splits (Tool #7) into three subproblems: (a) find the smallest integer $n_{\min}$ with $n_{\min}^3 \ge L$, (b) find the largest integer $n_{\max}$ with $n_{\max}^3 \le U$, (c) count the integers from $n_{\min}$ to $n_{\max}$ inclusive. For (a) the lower bound $257$ is small enough that Tool #6 (Guess & Check on $6^3, 7^3$) lands instantly. For (b) the upper bound $2^{18}+1$ looks scary, but Tool #9 (rewrite to an easier form) reveals the trick: $18 = 6 \times 3$, so $2^{18} = (2^6)^3 = 64^3$ — a perfect cube already sitting just below the upper bound.

Execute — Answer: E

#6 Guess and Check 6.EE.A.1 Step 1

Subproblem A — find the smallest cube at or above the lower bound.
Compute $2^8 + 1 = 256 + 1 = 257$, then test small cubes by guess-and-check.

$$6^3 = 216 < 257,\quad 7^3 = 343 \ge 257$$

💡 Evaluating $2^8$ and small whole-number cubes uses whole-number exponent expressions — a Grade 6 skill.

#7 Identify Subproblems 6.EE.B.5 Step 2

So the smallest perfect cube in the interval is $7^3 = 343$.
Any cube with base $n \le 6$ falls below $257$ and is excluded.

$$n_{\min} = 7$$

💡 Choosing $n_{\min} = 7$ is finding the smallest $n$ that makes the inequality $n^3 \ge 257$ true — a Grade 6 "solve an inequality by testing" move.

#9 Solve an Easier Related Problem 8.EE.A.1 Step 3

Subproblem B — find the largest cube at or below the upper bound.
Directly computing $2^{18}$ is painful, so rewrite using the exponent rule $(a^m)^n = a^{mn}$.
Since $18 = 6 \times 3$, we have $2^{18} = (2^6)^3 = 64^3$.
Therefore the upper bound is $64^3 + 1$, which is just $1$ more than the perfect cube $64^3$.

$$2^{18} = (2^6)^3 = 64^3,\quad \text{upper bound} = 64^3 + 1$$

💡 Rewriting $2^{18}$ as $(2^6)^3$ uses the integer-exponent property $(a^m)^n = a^{mn}$ — a Grade 8 idea that lets us skip a brute-force computation.

#7 Identify Subproblems 6.EE.B.5 Step 4

Compare $64^3$ and $65^3$ to the upper bound.
$64^3 \le 64^3 + 1$ (included), and $65^3 = (64+1)^3 > 64^3 + 1$ (excluded).
So the largest valid base is $64$.

$$n_{\max} = 64$$

💡 Picking the largest $n$ satisfying $n^3 \le 64^3 + 1$ is again testing the inequality on consecutive integers.

#7 Identify Subproblems 4.OA.A.3 Step 5

Subproblem C — count the integers $n = 7, 8, 9, \dots, 64$.
The count of consecutive integers from $a$ to $b$ inclusive is $b - a + 1$.
So the number of perfect cubes is $64 - 7 + 1 = 58$.
This matches choice $\textbf{(E)}$.

$$64 - 7 + 1 = 58 \;\Rightarrow\; \textbf{(E)}$$

💡 Counting how many whole numbers sit between two values, inclusive, is a Grade 4 multi-step word-problem move.

[1] #6 6.EE.A.1 Subproblem A — find the smallest cube at or above the lower bound. Compute $2^8

[2] #7 6.EE.B.5 So the smallest perfect cube in the interval is $7^3 = 343$. Any cube with base

[3] #9 8.EE.A.1 Subproblem B — find the largest cube at or below the upper bound. Directly compu

[4] #7 6.EE.B.5 Compare $64^3$ and $65^3$ to the upper bound. $64^3 \le 64^3 + 1$ (included), an

[5] #7 4.OA.A.3 Subproblem C — count the integers $n = 7, 8, 9, \dots, 64$. The count of consecu

Review

Reasonableness: The smallest cube is $7^3$ and the largest is $64^3$. The count is $64 - 7 + 1 = 58$, which is exactly choice (E). A quick sanity check on the "off-by-one": from $7$ to $64$ there are clearly more than $50$ integers (since $64 - 7 = 57$ gaps, plus one for the inclusive endpoint), ruling out (A)$=4$, (B)$=9$, (C)$=10$. The remaining trap is (D)$=57$, which is the mistake of forgetting the $+1$ — easy to spot since the problem says "inclusive".

Alternative: Tool #3 (Eliminate Possibilities) on the choices: (A) $4$, (B) $9$, (C) $10$ are all far too small — the interval spans from about $257$ to over $260{,}000$, which contains many cubes. That leaves only (D) $57$ and (E) $58$, separated exactly by the inclusive/exclusive endpoint question. Verifying that BOTH endpoints land on or next to cubes ($7^3 = 343$ is inside, and $64^3$ is exactly $1$ below the upper bound) confirms the inclusive count is (E) $58$.

CCSS standards used (min grade 8)

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers (Counting the integers from $7$ to $64$ inclusive using $64 - 7 + 1 = 58$.)
6.EE.A.1 Write and evaluate numerical expressions involving whole-number exponents (Evaluating $2^8 = 256$, $6^3 = 216$, and $7^3 = 343$ to locate the lower bound and the smallest cube in range.)
6.EE.B.5 Understand solving an equation or inequality as a process of finding values (Choosing the smallest $n$ with $n^3 \ge 257$ and the largest $n$ with $n^3 \le 64^3 + 1$ by testing successive integers.)
8.EE.A.1 Know and apply the properties of integer exponents (Rewriting $2^{18} = (2^6)^3 = 64^3$ via the exponent rule $(a^m)^n = a^{mn}$, which makes the upper bound a perfect cube.)

⭐ This AMC 8 problem only needs Grade 8 exponent rules — like $(2^6)^3 = 2^{18}$ — you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 8)

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