Primes & Prime Factorization • Topic 4 of 4

Perfect Squares & Cubes

A perfect square is an integer whose prime exponents are all even (36 = 2²·3²); a perfect cube has all exponents divisible by 3 (216 = 2³·3³). This "exponent parity" view answers a whole class of GRE questions instantly. To find the smallest multiplier that turns a number into a perfect square, supply whatever each prime needs to reach an even exponent — for 12 = 2²·3, the exponent of 3 is odd, so multiply by 3 to get 36. Two facts worth carrying: perfect squares end only in 0, 1, 4, 5, 6 or 9 (never 2, 3, 7, 8), and a perfect square has an odd number of factors. The GRE also likes the estimation angle — knowing 25² = 625 and 30² = 900 lets you place a square root between benchmarks without the calculator.

✅ Solved examples

1. Is 200 a perfect square?
200 = 2³·5². The exponent of 2 is 3, which is odd, so 200 is not a perfect square. (Its square root, ≈ 14.14, is not an integer.)
2. What is the smallest positive integer that multiplies 48 to give a perfect square?
48 = 2⁴·3. The exponent of 3 is odd; multiply by 3 to make it 2⁴·3² = 144 = 12². So the multiplier is 3.
3. What is the smallest positive integer that multiplies 108 to give a perfect cube?
108 = 2²·3³. To make all exponents multiples of 3, the 2² needs one more 2 (to reach 2³); 3³ is already fine. Multiply by 2: 216 = 6³. Multiplier = 2.
4. Quantitative Comparison. n has an odd number of positive factors. Quantity A: n. Quantity B: a perfect square. Which is greater / relationship?
An odd factor count means n IS a perfect square, so Quantity A is a perfect square — matching Quantity B's type. But the specific values are unknown, so the numerical relationship cannot be determined.

✏️ Practice — try these, take hints as needed

1. Is 441 a perfect square?
441 = 21².
Check 21 × 21.
Integer root?
Yes (21²)
2. Smallest multiplier to make 50 a perfect square?
50 = 2·5².
Exponent of 2 is odd.
Multiply by 2.
2
3. Smallest multiplier to make 72 a perfect cube?
72 = 2³·3².
Need 3³.
Multiply by 3.
3
4. Can a perfect square end in the digit 7?
Squares end in 0,1,4,5,6,9.
7 is not in the list.
So no.
No
5. Between which two consecutive integers does √150 lie?
12² = 144.
13² = 169.
144 < 150 < 169.
12 and 13

📝 Topic test — 8 questions

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