Competition · AMC preparation · step 4 of 4
AMC 8 · 2016 · #7
Grade 8 number-theoryPick an answer.
AMC 8 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Five answer choices and a yes/no test on each — Tool #3 (Eliminate) is the natural lead. The test itself comes from Tool #5: by checking small cases like 2², 2³, 2⁴, 2⁵, the pattern is clear — a power aⁿ (with a prime) is a perfect square exactly when n is even, because a²k = (a^k)². Tool #9 (Easier Problem) is what lets us trust that pattern: we verify it on tiny exponents before applying it to 2016–2020. The one trap is choice (D), where the base 4 is not prime; we rewrite 4 = 2² first so the exponent rule applies to a prime base.
Find the perfect-square test
Build the test from small cases: 2² and 2⁴ are squares but 2³, 2⁵ aren't — so a prime power pⁿ is a perfect square exactly when n is even.
Walking through n = 2, 3, 4, 5 on base 2 makes the even-exponent rule visible without any abstract proof — exactly the small-cases move.
6.EE.A.1Solve An Easier Related ProblemCheck choice A
(A) 1²⁰¹⁶ = 1 = 1², a perfect square — cross off (A).
Every power of 1 is just 1, and 1 is trivially 1².
6.EE.A.1Eliminate PossibilitiesCheck choice B
(B) base 2 is prime and exponent 2017 is odd, so 2²⁰¹⁷ is not a perfect square. The candidate answer; still check the other three.
Odd exponent on a prime base means you can't split the prime factors into two equal groups.
2²⁰¹⁷ is not a perfect square.
▸ Why?
A perfect square is a whole number multiplied by itself, and squaring a power of 2 doubles how many 2's it holds, so any perfect-square power of 2 must contain an even number of 2's — yet 2²⁰¹⁷ holds an odd number of them, 2017.
▸ Why?
2²⁰¹⁷ is the base 2 written out and multiplied together 2017 times, so it holds exactly 2017 twos, and 2017 is an odd count.
▸ Why?
Squaring 2^m is 2^m × 2^m, which sets m twos beside another m twos, doubling the run to 2m twos — an even count.
▸ Why?
Merging the two runs of twos into a single product and regrouping the factors leaves the product unchanged, so 2^m × 2^m truly holds 2m twos.
▸ Why?
The two identical factors of a perfect square let every 2 in one factor pair with a matching 2 in its twin, so the 2's come in complete pairs and their count is even; the odd total 2017 always leaves one 2 without a partner.
Check choice C
(C) base 3 is prime and exponent 2018 is even, so 3²⁰¹⁸ = (3¹⁰⁰⁹)² — a perfect square. Cross off (C).
Half the exponent gives the integer whose square it is.
6.EE.A.1Eliminate PossibilitiesCheck choice D
(D) trap: base 4 = 2² isn't prime, so 4²⁰¹⁹ = (2²)²⁰¹⁹ = 2⁴⁰³⁸ — 4038 is even, so it's (2²⁰¹⁹)². A square; cross off (D).
Whenever the base is itself a square, the answer is automatically a square no matter what the outer exponent is.
8.EE.A.1Eliminate PossibilitiesCheck choice E
(E) base 5 is prime and exponent 2020 is even, so 5²⁰²⁰ = (5¹⁰¹⁰)² — a perfect square. Only (B) survives.
Four of the five are squares; the only one left standing is (B).
6.EE.A.1Eliminate PossibilitiesA prime raised to an odd power can never be a perfect square — and that single Grade 8 exponent rule is enough to spot the odd-one-out among (A)–(E).
- Find the perfect-square test
- Check choice A
- Check choice B
- Check choice C
- Check choice D
- Check choice E
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