Pairing up prime factors

The idea

A number is a perfect square if its prime factors can be split into two identical groups (paired up); the product of the prime factors in one group is its square root.

Share the prime cards

A perfect square is a number times itself. Look at 324 = 2 × 2 × 3 × 3 × 3 × 3.

If 324 is a number times itself, what would each of the two equal groups of primes need?

• One 2 and two 3s
• Two 2s
• Four 3s

Two identical groups

Tap a card to move it between the pool and the groups. Make the groups identical. If a number cannot be split, press the button!

Pair the primes

A number is a perfect square if its prime factors can be split into two identical groups (paired up). The product of the primes in one group is its square root.

324 = (2 × 2) × (3 × 3) × (3 × 3). Every prime has a partner. One group is 2 × 3 × 3 = 18, so 324 = 18² and its square root is 18.

156 = 2 × 2 × 3 × 13. The 3 and the 13 have no partners, so 156 is not a perfect square.

Notes

A number is a perfect square if its prime factors split into two identical groups (pairs). The product of one group is the square root.

Check yourself

Which of these is a perfect square?

1764 = 2 × 2 × 3 × 3 × 7 × 7. Find the square root of 1764.

Answer: 42

One group is 2 × 3 × 7 = 42, so 1764 = 42 × 42 and its square root is 42.

3600 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5. Find its square root.

Answer: 60

One group: 2 × 2 × 3 × 5 = 60, so 3600 = 60² and its square root is 60.

• 150 = 2 × 3 × 5 × 5. 2 and 3 have no partners, so 150 cannot be split into two identical groups.
• 180 = 2 × 2 × 3 × 3 × 5. The 5 has no partner, so 180 cannot be split into two identical groups.
• 196 = 2 × 2 × 7 × 7 — correct. Yes! Each group is 2 × 7, so 196 = 14 × 14.
• 200 = 2 × 2 × 2 × 5 × 5. One 2 is left without a partner, so 200 is not a perfect square.