**Properties of Square Numbers**

- A number that ends in 2 , 3 , 7 or 8 is never a perfect square.
- The ones digit in the square of a number can be determined if the ones digit of the number is known. Look at the following table

- The number of zeroes at the end of a perfect square is always even e.g. 40
^{2}= 1600 , 200^{2}= 40000 - The square of an even number is always an even number and square of an odd number is always an odd number.
e.g. 12
^{2}= 144 , 23^{2}= 529 - If n is a perfect square then 2n can never be a perfect square. e.g. 100 is a perfect square , but 2 x 100 = 200 is not a perfect square number.
- The difference between the squares of the two consecutive numbers is equal to their sum or twice the smaller number plus 1. e.g.

^{2}- m

^{2}= [(m+1) + m] [(m+1) – m] = (2m +1)

- If a number is a square number , it has to be the sum of successive odd numbers starting from 1. e.g.

^{2}[Sum of first two odd numbers]

1 + 3 + 5 + 7 + 9 = 25 = 5

^{2}[ Sum of first five odd numbers]

- The square of a number , either negative or positive is always positive. e.g. (-3)
^{2}= (-3) x (-3) = 9 - The square of a natural number other than 1 is either a multiple of 3 or exceed the multiple of 3 by 1. Thus we can express square of a number (other than 1) as 3m or 3m+1 for some natural number m. e.g.

^{2}= 25 = 3 x 8 + 1

12

^{2}= 144 = 2 x 36

- The square of a natural number other than 1 is either a multiple of 4 or exceed the multiple of 4 by 1. Thus we can express square of a number (other than 1) as 4m or 4m+1 for some natural number m. e.g.

^{2}= 49 = 4 x 12 + 1

14

^{2}= 196 = 4 x 49
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ReplyDeleteThank you Madam Uma.

ReplyDeleteSome of these can be extended to broader statements. Like 2n is never a prime when n is a prime. Well, mn is a prime iff m and n are primes, right, so 2n is just a special case of this.

ReplyDeleteThanks for the thought provoking list!

nice post,..

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