Problem 7 · 2016 AMC 8
Medium
Number Theory
perfect-squarefactorization
Which of the following numbers is not a perfect square?
Show answer
Answer: B — 2^2017.
Show hints
Hint 1 of 3
A perfect square is something split into two identical halves. A power like 32018 is a perfect square exactly when you can break it into two equal piles — so what does that say about its exponent?
Still stuck? Show hint 2 →
Hint 2 of 3
Two tests catch every square here: (1) an EVEN exponent on any base means you can halve it, e.g. 32018 = (31009)2; (2) a base that's already a perfect square (like 4 = 22) stays a square no matter the exponent. Hunt for the ONE choice that passes neither.
Still stuck? Show hint 3 →
Hint 3 of 3
Watch the trap: 4 is not prime — rewrite it as 22 before judging its exponent, or you'll mis-count.
Show solution
Approach: rewrite each as a single prime power and check if the exponent is even
- Reduce each base to a prime: 12016 = 1 = a square trivially; 32018 and 52020 have EVEN exponents on a prime, so they split into equal halves — squares.
- 42019 = (22)2019 = 24038, an even exponent — also a square.
- 22017 is a prime with an ODD exponent (2017): you can't split it into two equal prime-power piles, so it's NOT a perfect square.
- Why this transfers: a number is a perfect square exactly when EVERY prime in its factorization has an even exponent — reduce to primes first, then the parity of each exponent settles it.
Mark:
· log in to save