Prime factorisation: 75 = 3 × 5². Not all pairs → not perfect square.

Is 75 a Perfect Square? How to Check

Question

Is 75 a perfect square? Use prime factorisation to verify your answer.

Solution — Step by Step

We break 75 down into its prime factors. Divide by the smallest prime that works:

75 = 3 \times 25 = 3 \times 5 \times 5

So the prime factorisation is $75 = 3^1 \times 5^2$ .

For a number to be a perfect square, every prime factor must appear an even number of times — that’s the rule.

Prime	Power	Paired?
3	1	No (odd)
5	2	Yes

Factor 5 pairs up nicely. Factor 3 is left alone with no partner.

Since 3 has an odd power (power = 1), we cannot form a complete pair for it.

A perfect square needs every prime to appear in pairs — 75 fails this test.

75 is NOT a perfect square.

The nearest perfect squares around 75 are $8^2 = 64$ and $9^2 = 81$ .

Why This Works

When we write a perfect square like $36 = 2^2 \times 3^2$ , we can split it evenly into two identical groups: $36 = (2 \times 3) \times (2 \times 3) = 6 \times 6$ . Every prime pairs up perfectly.

With 75, the single 3 has no pair. We’d need one more factor of 3 to make $75 \times 3 = 225 = 15^2$ , which IS a perfect square.

This pairing idea is the entire logic behind the NCERT method for finding the smallest multiplier or divisor to make a number a perfect square — a question type that appears almost every year in Class 8 boards.

Alternative Method

Estimate the square root and check:

We know $8^2 = 64$ and $9^2 = 81$ . Since $64 < 75 < 81$ , the square root of 75 falls strictly between 8 and 9.

\sqrt{75} \approx 8.66...

Since $\sqrt{75}$ is not a whole number, 75 is not a perfect square. This quick estimation method is handy for MCQs when you need a fast answer without full factorisation.

For any number between two consecutive perfect squares $n^2$ and $(n+1)^2$ , you can immediately say it is NOT a perfect square. No calculation needed — just recognise where it falls on the number line.

Common Mistake

A very common error: students see $75 = 3 \times 25$ and think “25 is a perfect square, so 75 should be related to one.” Not quite. A factor of 75 being a perfect square does NOT make 75 itself a perfect square. The test applies to the complete prime factorisation of the number, not just one factor. Always check ALL prime factors and their powers.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

Try These Next