Let’s look at the square of numbers from 1 to 50
Number | Square |
1 | 1 |
2 | 4 |
3 | 9 |
4 | 16 |
5 | 25 |
6 | 36 |
7 | 49 |
8 | 64 |
9 | 81 |
10 | 100 |
11 | 121 |
12 | 144 |
13 | 169 |
14 | 196 |
15 | 225 |
16 | 256 |
17 | 289 |
18 | 324 |
19 | 361 |
20 | 400 |
21 | 441 |
22 | 484 |
23 | 529 |
24 | 576 |
25 | 625 |
30 | 900 |
35 | 1225 |
40 | 1600 |
45 | 2025 |
50 | 2500 |
Let’s see some pattern in it, and find properties of square numbers
A Square number can only end with digits 0, 1, 4, 5, 6, 9
Example :
1, 4, 9, 16, 25, 36, 49, 64, 81, 100 are square numbers.
But, 28, 97 will not be a perfect square
Number of zeroes at the end of a perfect square is always even
Example :
2500 is a perfect square
100 is a perfect square
But, 80 is not a perfect square
4000 is also not a perfect square
Square of even numbers are always even,
Square of odd numbers are always odd
Example :
Square of 2 is 4,
Square of 6 is 36
And
Square of 7 is 49
Square of 9 is 81
If a number has 1 or 9 in its unit place,
its square ends with 1
Example:
Number |
Square |
1 |
1 |
9 |
181 |
11 |
121 |
19 |
361 |
If a number has 4 or 6 in its unit place,
its square ends with 6
Example:
Number |
Square |
4 |
16 |
6 |
36 |
14 |
196 |
16 |
256 |
Unit digit of square of any number will be the unit digit of the square of its last digit
Example:
For number 29
Unit digit of 29 ^{ 2 } = Unit digit of 9 ^{ 2 }
= Unit digit of 81
= 1
For number 76
Unit digit of 76 ^{ 2 } = Unit digit of 6 ^{ 2 }
= Unit digit of 36
= 6
A perfect square always leaves remainder 0 or 1 when divided by 4
Perfect Square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100,..
- 1 on divided by 4 leaves remainder 1
- 4 on divided by 4 leaves remainder 1
- 9 on divided by 4 leaves remainder 0
- 16 on divided by 4 leaves remainder 1
- 25 on divided by 4 leaves remainder 1
- 36 on divided by 4 leaves remainder 0
So, if we need to check whether 98 is a perfect square,
we check its remainder when divided by 4
98 on divided by 4 leaves remainder 2
So, it is not a perfect square
Square number is sum of consecutive odd numbers
Check explanation here
For any Natural number m greater than 1 (2m, m ^{ 2 } - 1, m ^{ 2 } +1) is a pythagoras triplet
Check explanation here