Perfect Squares • Topic 2 of 2

Number of Factors Test

The factor count gives a clean, calculation-proof test for perfect squares. Write the prime factorisation N = p^a · q^b · r^c …; the total number of factors is (a+1)(b+1)(c+1)…. A number is a perfect square exactly when every prime exponent is even — because pairing primes into two equal halves needs even powers. A beautiful consequence: a perfect square is the only kind of number with an ODD number of factors. Every factor d pairs with N/d, and these pairings cover all factors in twos — except when d = N/d = √N, which happens only for a square, leaving one unpaired factor and an odd total. CAT uses this two ways. Forward: "how many of 1–100 have an odd number of factors?" Answer: the perfect squares, 1, 4, …, 100 — that is 10. Backward: to make N a perfect square, multiply by the product of exactly those primes whose exponent is currently odd, lifting each odd power to the next even power.

✅ Solved examples

1. How many integers from 1 to 100 have an odd number of factors?
Only perfect squares have an odd factor count. The squares ≤ 100 are 1², 2², …, 10² ⇒ 10 numbers.
2. Is 3,600 a perfect square? Use the exponent test.
3600 = 2^4 · 3^2 · 5^2. Every exponent (4, 2, 2) is even ⇒ 3600 is a perfect square (60²). Factor count = 5×3×3 = 45, which is odd, confirming it.
3. What is the smallest positive integer by which 1,176 must be multiplied to make a perfect square?
1176 = 2^3 · 3^1 · 7^2. Odd exponents are on 2 and 3. Multiply by 2 × 3 = 6 ⇒ 1176 × 6 = 7056 = 84².
4. N = 2^6 · 5^4 · 7^3. How many of its factors are themselves perfect squares?
A square factor uses even exponents only. Choices: for 2 (0,2,4,6) = 4 ways; for 5 (0,2,4) = 3 ways; for 7 (0,2) = 2 ways. Total = 4×3×2 = 24 square factors.

✏️ Practice — try these, take hints as needed

1. A number has an odd number of factors. What must it be?
Factors pair up as d and N/d.
Odd count needs one self-pair.
That means √N is an integer.
A perfect square
2. Is 1,225 a perfect square? Factorise.
1225 = 5^2 · 7^2.
All exponents even.
Take the square root.
Yes, 35²
3. Smallest multiplier to make 588 a perfect square?
588 = 2^2 · 3 · 7^2.
Only 3 has an odd power.
Multiply by 3.
3
4. Smallest number to DIVIDE 2,028 by to get a perfect square? (2028 = 2^2 · 3 · 13^2)
Find the odd-power prime.
Exponent of 3 is 1.
Divide out that prime.
3
5. How many factors of 36 are perfect squares? (36 = 2^2 · 3^2)
Square factors use even exponents.
2: powers 0 or 2 (2 ways); 3: 0 or 2 (2 ways).
2 × 2.
4 (namely 1, 4, 9, 36)

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Square Properties Take the chapter test →