Properties and patterns of square numbers |
The square of an even number is always an even number. For example: the square of 4 is 16 which is an even number. |
The square of an odd number is always an odd number. For example: the square of 15 is 225 which is an odd number. |
A number ending in 2, 3, 7 or 8 is never a perfect square. For example: the numbers 42, 33, 57 and 68 ends in 2, 3, 7 and 8 respectively. So, none of them is a perfect square. |
If we know the unit's digit of a number, we can determine the unit's digit of its square.
Unit's digit of number |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
Unit's digit of square |
0 |
1 |
4 |
9 |
6 |
5 |
6 |
9 |
4 |
1 |
|
The number of zeros at the end of a perfect square is always even. If a number ends in a zero, its square ends in a double zero. |
A perfect square leaves a remainder 0 or 1 on division by 3. For example: when 16 is divisible by 3 we get remainder as 1 and similarly, when 9 is divisible by 3 we get remainder as 0. |
Square of a number: If a number is multiplied by itself, then the result is called the square of that number. In other words, the square of a number is that number raised to the power 2.
Ex: 12 × 12 = 122 = 144
So 144 is the square of 12.
Mathematically, we can say that: if 'm' and 'n' are two numbers such that m * m = n, then 'n' is the 'square of m' and called as 'square number'. For example, in the product 8 * 8 = 64, 64 is the square number.
Example: |
How many numbers lie between the squares of 10 and 11? |
Solution: |
102 = 100
112 = 121
The numbers between 121 and 100 are: 101, 102, . . ., 120.
∴ 20 numbers are there between 102 and 112. |
Perfect squares: A natural number is called a perfect square, if it is square of some other natural number.
Examples: 4 (2 × 2), 25 (5 × 5), 64 (8 × 8), . . .
Note: A given number is a perfect square, if it can be expressed as the product of exact number of pairs of equal factors.
If there are 'n' digits in given number, the square will have either 2n or 2n – 1 digits. This condition is true only for numbers above '3', because the square of 2 = 4 and the square of 3 = 9.
For example, 52 = 25. Here '5' is a single digit number (n = 1) whereas 25 is two digits number (2n = 2 × 1 = 2).
Similarly, 122 = 144. Here '12' is a two digits number (n = 2) where as 144 is three digits number (2n – 1 = 2 × 2 – 1 = 3).