Properties of Perfect Squares

The properties of perfect squares are explained here in each property with examples.

Property 1:

Numbers ending in 2, 3, 7 or 8 is never a perfect square but on the other hand, all the numbers ending in 1, 4, 5, 6, 9, 0 are not square numbers.

For example:

The numbers 10, 82, 93, 187, 248 end in 0, 2, 3, 7, 8 respectively.

So, none of them is a perfect square.

Property 2:

A number ending in an odd number of zeros is never a perfect square.

For example:

The numbers 160, 4000, 900000 end in one zero, three zeros and five zeros respectively.

So, none of them is a perfect square.

Property 3:

The square of an even number is always even.

For example:

2² = 4, 4² = 16, 6² = 36, 8² = 64, etc.

Property 4:

The square of an odd number is always odd.

For example:

1² = 1, 3² = 9, 5² = 25, 7² = 49, 9² = 81, etc.

Property 5:

The square of a proper fraction is smaller than the fraction.

For example:

(2/3)² = (2/3 × 2/3) = 4/9 and 4/9 < 2/3, since (4 × 3) < (9 × 2).

Property 6:

For every natural number n, we have

(n + 1)² - n² = (n + 1 + n)(n + 1 - n) = {(n + 1) + n}.

Therefore, {(n + 1)² - n²} = {(n + 1) + n}.

For example:

(i) {1 + 3 + 5 + 7 + 9} = sum of first 5 odd numbers = 5²

(ii) {1 + 3 + 5 + 7 + 9 + 11 + 13 + 15} = sum of first 8 odd numbers = 8²

Property 7:

For every natural number n, we have

sum of the first n odd numbers = n²

For example:

(i) {1 + 3 + 5 + 7 + 9} = sum of first 5 odd numbers = 5²

(ii) {1 + 3 + 5 + 7 + 9 + 11 + 13 + 15} = sum of first 8 odd numbers = 8²

Property 8 (Pythagorean Triplets):

Three natural numbers m, n, p are said to form a Pythagorean triplet (m, n, p) if (m² + n²) = p².

Note:

For every natural number m > 1, we have (2m, m² – 1, m² + 1) as a Pythagorean triplet.

For example:

(i) Putting m = 4 in (2m, m² – 1, m² + 1) we get (8, 15, 17) as a Pythagorean triplet.

(ii) Putting m = 5 in (2m, m² – 1, m² + 1) we get (10, 24, 26) as a Pythagorean triplet.

Solved examples on the properties of perfect squares;

1. Without adding, find the sum (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17).

Solution:

(1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17) = sum of first 9 odd numbers = 9² = 81

2. Express 49 as the sum of seven odd numbers.

Solution:

49 = 7² = sum of first seven odd numbers

= (1 + 3 + 5 + 7 + 9 + 11 + 13).

3. Find the Pythagorean triplet whose smallest member is 12.

Solution:

For every natural number m > 1. (2m, m² – 1, m² + 1) is a Pythagorean triplet.

Putting 2m = 12, i.e., m = 6, we get the triplet (12, 35, 37).

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