Properties of A Perfect Square
Please look at the table of some squares given below. You will see interesting patterns here.
Square of number 0 = 0
| Number |
Square |
Number |
Square |
Number |
Square |
| 1 |
1 |
11 |
121 |
21 |
441 |
| 2 |
4 |
12 |
144 |
22 |
484 |
| 3 |
9 |
13 |
169 |
23 |
529 |
| 4 |
16 |
14 |
196 |
24 |
576 |
| 5 |
25 |
15 |
225 |
25 |
625 |
| 6 |
36 |
16 |
256 |
26 |
676 |
| 7 |
49 |
17 |
289 |
27 |
729 |
| 8 |
64 |
18 |
324 |
28 |
784 |
| 9 |
81 |
19 |
361 |
29 |
841 |
| 10 |
100 |
20 |
400 |
30 |
900 |
PROPERTY 1:
Look at the table above. Observe that none of the column of squares ends with 2, 3, 7, or 8. Therefore we can say that a number endings with 2, 3, 7, 8 can never be a perfect square.
Thus 228257 , 132457 , 189678 , 84453 are not perfect squares.
PROPERTY 2:
Look at the last row of the table shown above. We notice that all the perfect squares are ending with an even number or 0. In the first line e.g. 10² = 100 , 20² = 400, 30² =900,
Similarly 80² = 6400, 50² = 2500.
We can prime factorise and see that 10, 20, 30, etc . are not perfect squares. Thus a number ending with odd number of zeros can never be a perfect square.
