Giving the property of square numbers, identified which are not perfect square
58000, 1234567, 88888, 333222
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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
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