A Square and A Cube
What the last digit tells you
Find out what the last digit of a number can — and cannot — tell you about whether it is a square.
Write out the squares of the first 30 natural numbers and look only at the last digit: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, …
Every one ends in 0, 1, 4, 5, 6 or 9. None ends in 2, 3, 7 or 8.
Be careful what that tells you. A number ends in 6: is it a square? 16 and 36 are, but 26 is not. So the last digit can never tell you that a number is a square.
What it can do is rule one out. If a number ends in 2, 3, 7 or 8, it is not a square: a one-second test worth having.
You don't need to square a whole number to find its last digit: just square its last digit. 3142189² must end in 1, because 9² = 81. That's why the last digits pair up as 'friends':
| Number ends in | Its square ends in | Examples |
|---|---|---|
| 1 or 9 | 1 | 1² = 1, 9² = 81, 11² = 121, 19², 21², 29² |
| 2 or 8 | 4 | 2² = 4, 8² = 64 |
| 3 or 7 | 9 | 3² = 9, 7² = 49 |
| 4 or 6 | 6 | 4² = 16, 6² = 36, 14² = 196, 16² = 256, 24² = 576, 26² = 676 |
| 5 | 5 | 5² = 25 |
| 0 | 0 | 10² = 100 |
Now the zeros: 10² = 100 (one zero becomes two), 20² = 400, 100² = 10000 (two zeros become four), 200² = 40000, 700² = 490000, 900² = 810000. The zeros at the end of the square are double the zeros at the end of the number. So a number ending in 3 zeros has a square ending in 6.
That gives another quick test: a square can only end in an even number of zeros. And one rule about even and odd: an even number's square is even, and an odd number's square is odd.
Practise this lesson
Questions that get harder as you improve, a Tutor that helps without giving the answer away, and progress you can see. Free for every learner.
No account, no email address. Opens the real product.