AMC 10 · 2021 · #4

Grade 8 algebra
Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In
exponentsprime-factorizationexponential-functionmental-arithmetic easier-related-problemidentify-subproblems ↑ Prerequisites: exponents
📏 Short solution 💡 2 insights
Problem
A single number is defined as eight raised to the two thousand twenty-second power. Divide it by four and decide which of the five listed powers — written in base four, base two, or base eight — is exactly equal to the result.

Pick an answer.

(A)
$4^{1010}$
(B)
$2^{2022}$
(C)
$8^{2018}$
(D)
$4^{3031}$
(E)
$4^{3032}$
How to solve
Strategy Organize Information in More Ways

Nothing here is hard to compute; the difficulty is that the same number is wearing three different costumes — base 8, base 4, base 2 — and costumes cannot be compared. Tool #15 (Organize Information in More Ways) fixes that by rewriting every quantity in the one base they all share, namely 2; once everything is 2^something, the entire problem collapses to comparing two exponents. Tool #9 (Solve an Easier Related Problem) guards the machinery: swapping 2022 for a tiny exponent like 2 makes the whole thing a 64÷4 arithmetic check, which confirms the rule before it is trusted at scale. Tool #3 (Eliminate Possibilities) finishes the job — with all five choices translated into base 2, exactly one exponent can match, and the other four are ruled out on sight.

1STEP 1

Pick the shared base

Take two as the shared base.

8=2³, 4=2²
2STEP 2

Rewrite n as a power of 2

Rewrite it as a power of two.

n=8²⁰²²=(2³)²⁰²²=2³ · 2022=2⁶⁰⁶⁶
3STEP 3

Divide by 4, remove two 2s

Dividing by four drops the exponent by two.

n/4=2⁶⁰⁶⁶/2²=2⁶⁰⁶⁶⁻²=2⁶⁰⁶⁴
4STEP 4

Test the rule on a small case

Test the rule on a small case.

8²=2⁶, 2⁶/2²=2⁴=16=4²
5STEP 5

Translate the choices and match

Translating the choices gives four to the three thousand thirty-second.

4³⁰³²=(2²)³⁰³²=2⁶⁰⁶⁴=n/4 → (E)
Answer
4³⁰³²
Dividing by 4 barely dents a number as large as 8²⁰²², so n/4 must still be enormous — its base-2 exponent should sit just 2 below 6066. Choices (A) 2²⁰²⁰ and (B) 2²⁰²² are not merely smaller, they are smaller by thousands of doublings, which no single division by 4 could cause. Choice (C) 8²⁰¹⁸=2⁶⁰⁵⁴ is the trap of subtracting 4 from the exponent of the base-8 form, which actually divides by 8⁴=4096. Choice (D) 2⁶⁰⁶² is the off-by-one trap of removing four 2s instead of two, i.e. dividing by 16. Only (E) =2⁶⁰⁶⁴ is exactly two doublings below n, which is what dividing by 4 does.
💡Key takeaway

When powers with different bases are being compared, rewrite them all in the one base they share — then the whole problem is just adding and subtracting exponents.

  • Pick the shared base
  • Rewrite n as a power of 2
  • Divide by 4, remove two 2s
  • Test the rule on a small case
  • Translate the choices and match

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