48 = 2⁴ × 3. Need one more 3 → multiply by 3.

Find the Smallest Number to Multiply 48 to Make a Perfect Square

Question

What is the smallest number by which 48 must be multiplied so that the product is a perfect square?

Solution — Step by Step

Break 48 down to its prime factors:

48 = 2 \times 24 = 2 \times 2 \times 12 = 2 \times 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 2 \times 3

So $48 = 2^4 \times 3^1$ .

For a perfect square, every prime factor must appear an even number of times in the factorisation. We check the exponents:

$2^4$ → exponent is 4, which is even ✓
$3^1$ → exponent is 1, which is odd ✗

The prime 3 is unpaired. That’s our problem.

To make the exponent of 3 even, we need one more factor of 3. Multiplying 48 by 3 gives:

48 \times 3 = 144 = 2^4 \times 3^2

Now every prime has an even exponent.

\sqrt{144} = \sqrt{2^4 \times 3^2} = 2^2 \times 3 = 4 \times 3 = 12

We get a whole number — confirmed perfect square.

The smallest number is 3.

Why This Works

A perfect square, when prime factorised, always has even exponents on every prime. Think of it this way: when you take the square root, you’re dividing each exponent by 2. If any exponent is odd, that division leaves a remainder — meaning the square root isn’t a whole number.

So the strategy is simple: find every prime with an odd exponent, and multiply in exactly enough copies of those primes to make the exponents even. We’re doing the minimum work possible — that’s why the answer is always the product of primes-with-odd-exponents, each taken to the power of 1.

Here, only $3^1$ has an odd exponent, so we multiply by exactly $3^1 = 3$ .

Alternative Method

You can also spot this by grouping the factors in pairs:

48 = \underbrace{2 \times 2}_{\text{pair}} \times \underbrace{2 \times 2}_{\text{pair}} \times 3

Every factor needs a pair to be part of a perfect square. The two 2-pairs are fine. But that single 3 has no partner. Multiply by 3 to give it one:

48 \times 3 = (2 \times 2) \times (2 \times 2) \times (3 \times 3) = 4 \times 4 \times 9 = 144

This “pairing” visual is often faster in your head during an exam.

Once you find the unpaired primes, the answer is just their product. If unpaired primes were $3$ and $5$ , you’d multiply by $3 \times 5 = 15$ . No need for any extra computation.

Common Mistake

Students often multiply by the entire prime factorisation again — writing $48 \times 48 = 2304$ and claiming that’s the answer. Yes, $2304$ is a perfect square, but $48$ is not the smallest multiplier. Always check which specific primes have odd exponents and multiply only by those. The answer must be the minimum, not just any working value.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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