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
2^{2}= 4 = 3 x 1 + 1
1
3^{2}= 9 = 3 x 3 + 0
0
4^{2}= 16 = 3 x 5 + 1
1
5^{2}= 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