General Properties of Squares
General Properties of Squares
The square numbers are the values which are produced when we multiply a number by itself. Some of the properties are:
Property 1: The square of a natural number other than one is either a multiple of 3 or exceeds a multiple of 3 by 1.
In other words, a perfect square leaves remainder 0 or 1 on division by 3.
Verification: The above property can be easily verified from the following computations:
| Square number | Remainder when divided by 3 |
| 1 | |
| 0 | |
| 1 | |
| 1 | |
| 0 | |
| 1 | |
| 1 |
Application: If a number when divided by 3 leaves remainder 2, then it is not a perfect square.
Property 2: The square of a natural number other than one is either a multiple of 4 or exceeds a multiple of 4 by 1.
In other words, a perfect square leaves remainder 0 or 1 on division by 4.
Property 3: The square of a natural number n is equal to the sum of first n odd natural numbers.
We have,
Sum of first 1 odd natural number,
Sum of first 2 odd natural numbers,
Sum of first 3 odd natural numbers.
Sum of first 4 odd natural numbers.
Property 4 : The sum of first n odd natural numbers is 
i.e.
Find the sum of first 5 odd natural numbers using properties of square. | |||
| Right Option : D | |||
| View Explanation | |||
The sum 1 + 3 + 5 + 7 + 9 is _________________ .
| |||
| Right Option : A | |||
| View Explanation | |||
144 can be express as the sum of first ___________________ consecutive odd numbers. | |||
| Right Option : C | |||
| View Explanation | |||
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