Last digits of squares

The idea

Perfect squares end only in 0, 1, 4, 5, 6 or 9, with an even number of zeros at the end; a number ending in 2, 3, 7 or 8 is never a square, but other endings cannot confirm one.

Look at the ends

The squares of the first few numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, …

Look only at the last digit of each one.

Can a perfect square end in 7?

• Yes
• No
• Only a very big one

Fill the boxes

Tap a number to square it. Its last digit drops into a box. When you are ready, tap Square all 30.

Only six endings

Perfect squares end only in 0, 1, 4, 5, 6 or 9, and always with an even number of zeros. A number ending in 2, 3, 7 or 8 is never a square.

The other endings do not prove it is a square. 16 and 36 are squares ending in 6, but 26 is not.

• Number | Its square | Zeros at the end
• 10 | 100 | 2
• 20 | 400 | 2
• 100 | 10000 | 4
• 200 | 40000 | 4

Notes

Perfect squares end only in 0, 1, 4, 5, 6 or 9 (and in an even number of zeros). Ending in 2, 3, 7 or 8 means not a square; the other endings cannot confirm one.

Check yourself

Which of these can not be a perfect square?

Could 4000 be a perfect square?

The number 26 ends in 6, like the squares 16 and 36. Is 26 a square?

How many different digits can a perfect square end in?

Answer: 6

0, 1, 4, 5, 6 and 9: six different endings.

• 4096. 4096 ends in 6, so it might be a square. It is: 64 × 64.
• 3253 — correct. Yes! It ends in 3. No square ends in 2, 3, 7 or 8.
• 1296. 1296 ends in 6, so it might be a square. It is: 36 × 36.
• 2025. 2025 ends in 5, so it might be a square. It is: 45 × 45.
• Yes, it ends in 0. Ending in 0 is allowed, but squares end in an even number of zeros.
• No, it has 3 zeros at the end — correct. Yes! Squares end in 2, 4, 6, … zeros. 4000 has three, an odd number.
• Yes, it ends in 6. Ending in 6 only means it might be a square.
• No — 5 × 5 = 25 and 6 × 6 = 36, so 26 is in between — correct. Yes! A good ending does not prove a square. We must still check.