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 |
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
• 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
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