Some Properties Of The Squares of Natural Numbers are as follows –
a) The squares of even numbers are even, while the squares of odd numbers are odd.
b) Perfect squares can never end with the digits 2, 3, 7 & 8, also they cannot have an odd number of zeros at the end. For example 342, 5013, 117, 3568, 25000 are not perfect squares.
c) The square of a natural number (other than 1) is –
1) Divisible by 3 or leaves the remainder 1 when divided by 3.
2) Divisible by 4 or leaves the remainder 1 when divided by 4.
Example - 15² = 225 = 3 X 75 = 4 X 56 + 1
16² = 256 = 3 X 85 + 1 = 4 X 64
d) The difference of the squares of two consecutive natural numbers is always odd and is equal to the sum of those consecutive natural numbers.
Example - 7² - 6² = 49 – 36 = 13 = 7 + 6
14² - 13² = 196 – 169 = 27 = 14 + 13
10² - 9² = 100 – 81 = 19 = 10 + 9
20² - 19² = 400 – 361 = 39 = 20 + 19
15² - 14² = 225 – 196 = 29 = 15 + 14