Properties of Square Numbers

In this section we will discuss properties of square numbers.

Property 1: A number having 2, 3, 7 or 8 at unit’s place is never a perfect square. In other words, no square number ends in 2, 3, 7 or 8.

Example: None of the numbers 152, 7693, 14357, 88888, 798328 is a perfect square because the unit digit of each number ends with 2,3,7 or 8
Property 2: The number of zeros at the end of a perfect square is always even. In other words, a number ending in an odd number of zeros is never a perfect square.
Example : 2500 is a perfect square as number of zeros are 2(even) and 25000 is not a perfect square as the number of zeros are 3 (odd).
Property 3: Squares of even numbers are always even numbers and square of odd numbers are always odd.

Example : 12 2 = 12 x 12 = 144. (both are even numbers)
19
2 = 19 x 19 = 361 (both are odd numbers)

Property 4: 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.
Square number Remainder when divided by 3
22= 4 = 3 x 1 + 1 1
32= 9 = 3 x 3 + 0 0
42= 16 = 3 x 5 + 1 1
52= 25 = 3 x 8 + 1 1

Example: 635,98,122 are not perfect squares as they leaves remainder 2 when divided by 3.
Property 5: The Square of a natural number other than one is either a multiple of 4 or exceeds a multiple of 4 by 1.
Example : 67,146,10003 are not perfect squares as they leave remainder 3,2,3 respectively when divided by 4.
Property 6: The unit’s digit of the square of a natural number is the unit’s digit of the square of the digit at unit’s place of the given natural number.
Example :
1) Unit digit of square of 146.
Solution : Unit digit of 6 2 = 36 and the unit digit of 36 is 6, so the unit digit of square of 146 is 6.

2) Unit digit of square of 321.
Solution : Unit digit of 1 2 = 1, so the unit digit of square of 321 is 1.

Property 7: There are n natural numbers p and q such that p 2 = 2q 2 .
Property 8: For every natural number n,
(n + 1)
2 - n 2 = ( n + 1) + n.
Properties of square numbers 9: The square of a number n is equal to the sum of first n odd natural numbers.
1
2 = 1
2
2 = 1 + 3
3
2 = 1 + 3 + 5
4
2 = 1 + 3 + 5 + 7 and so on.
Properties of square numbers 10: For any natural number m greater than 1,
(2m, m
2 - 1, m 2 + 1) is a Pythagorean triplet.
Squares and Square roots

Introduction of Squares and Square Roots
Perfect Squares or not
Properties of Square Numbers
Short cut method to find squares
Introduction of Square Roots
Properties of Square Roots
Square root by Prime factorization method
Square root by long division method
Square root of rational numbers
Square root of Decimals
Square root by estimation method

From squares and square roots to Exponents

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