Problem
Perfect squares from even prime exponents
A perfect square comes from multiplying a whole number by itself. Five numbers are given: 236, 441, 853, 576, 2904. Decide without multiplying candidates one by one. Pick out every perfect square.
Reasoning · Grade 5-1 Applications of Prime Factorization
Your answer
Watch out Some paired primes are not enough: a single odd count spoils it.
How to solve
Strategy Make a Systematic List — First I look at the three examples for a pattern that survives when the numbers get big. Writing 4 = 2 × 2, 25 = 5 × 5 and 121 = 11 × 11 in prime form shows every prime turning up an even number of times, and that is the test I will use. Then I make a systematic list: break each of the five candidates into primes, one row per candidate, and count how many times each prime appears. Anything with a prime appearing an odd number of times is out; anything where every count is even gets confirmed by actually multiplying the two halves back together.
1STEP 1
Turn the definition into a test on prime factors
In a square every prime factor comes in pairs.
k = p × q → k × k = p × q × p × q = p² × q²
Doubling a list doubles every count, and doubled counts are always even. Reading the pattern off the three tiny examples means you never have to guess which whole number to try.
6.EE.A.1Look For A PatternA number is a perfect square exactly when every prime in its factorization appears an even number of times.
Why?
A number breaks into primes in only one way, so squaring it simply doubles how many times each prime appears.
🧱Every number has one prime factorizationKeep splitting a number into factors and you always land on the same bag of primes, no matter which split you tried first.
Why?
A square is a number multiplied by an equal copy of itself, so its primes come in matched pairs with none left over.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
Candidate (1): factor 236
236 has a lone 59, so no.
236 = 2 × 2 × 59 = 2² × 59 → not a perfect square
One lonely prime is enough to sink a number, so you can stop the moment you spot an odd count.
4.OA.B.4Make A Systematic List3STEP 3
Candidate (2): factor 441
441 has two 3s and two 7s: 21 squared.
441 = 3 × 3 × 7 × 7 = 3² × 7² = (3 × 7)×(3 × 7) = 21 × 21
The digit-sum test for 3 finds the first factor without any trial division, and after that the numbers are small enough to recognise on sight.
6.EE.A.1Make A Systematic List4STEP 4
Candidate (3): rule out 853 by its last digit
853 ends in 3, so no.
last digits of squares ∈ {0,1,4,5,6,9}, 853 ends in 3
The ones digit of a product depends only on the ones digits being multiplied, so a ten-case check settles the ones digit of every square there will ever be.
4.OA.B.4Look For A Pattern5STEP 5
Candidate (4): factor 576
576 pairs up completely: 24 squared.
576 = 2⁶ × 3² = (2×2×2×3)×(2×2×2×3) = 24 × 24
Halving over and over needs nothing beyond division by 2, and it strips a number like 576 down to primes in a few tidy lines.
6.EE.A.1Make A Systematic List6STEP 6
Candidate (5): factor 2904
2904 has odd counts of 2 and 3, so no.
2904 = 2³ × 3 × 11² → not a perfect square
It is tempting to see the pair of 11s and call it a square, but the test is about every prime at once — the stray 2 and the stray 3 spoil it.
4.OA.B.4Make A Systematic List7STEP 7
Multiply the two winners back out
Multiplying back confirms only 441 and 576.
21 × 21 = 441 and 24 × 24 = 576
A factor tree can go wrong at any branch, so multiplying the answer back out is the one check that catches every kind of slip at once.
5.NBT.B.5Guess And CheckAnswer
441, 576
21 × 21 = 441, 24 × 24 = 576
The sizes line up. 441 sits between 20 × 20 = 400 and 22 × 22 = 484, and 21 × 21 is the only whole-number square in that gap, so 21 is the right size; 576 sits between 23 × 23 = 529 and 25 × 25 = 625, and 24 × 24 fits exactly. The three rejects fail a size check too: 236 lies strictly between 15² = 225 and 16² = 256, 853 between 29² = 841 and 30² = 900, and 2904 between 53² = 2809 and 54² = 2916 — each is caught between two consecutive squares, so no whole number can square to it. The last-digit rule agrees as well: 441 ends in 1 and 576 ends in 6, both allowed, while 853 ends in 3, which no square ever does.
Takeaway
Break a number into primes: if every prime shows up an even number of times you can deal them into two matching piles, and each pile is the number that squares to it.
- Turn the definition into a test on prime factors
- Candidate (1): factor 236
- Candidate (2): factor 441
- Candidate (3): rule out 853 by its last digit
- Candidate (4): factor 576
- Candidate (5): factor 2904
- Multiply the two winners back out