Perfect Square or Square Number

What are called perfect square or square number?

Natural numbers that are squares of other natural numbers are called perfect square or square number.

For example;

We know that; 1 = 1²; 4 = 2²; 9 = 3²; 16 = 4²; 25 = 5² and so on.

Thus 1, 4, 9, 16, 25, etc., are perfect squares.

To find out if the given number is a perfect square:

If the prime factors of a number are grouped in pairs of equal factors, then that number is called a perfect square. Or, in other words if a perfect square number is always expressible as the product of pairs of equal factors.

1. Find out if the following numbers are perfect squares:

(i) 144 (ii) 90 (iii) 180

(i) 144

Resolving 144 into prime factors, we get

$Prime Factors$

144 = 2 × 2 × 2 × 2 × 3 × 3

(grouping the factors into the pairs of equal factors)

Therefore, 144 is a perfect square.

(ii) 90

Resolving 90 into prime factors, we get

$Prime Factors$

90 = 2 × 3 × 3 × 5

(Here 3 is grouped in pairs of equal factors and 2 and 5 are not grouped in pairs of equal factors)

Therefore, 90 is not a perfect square.

(iii) 180

Resolving 180 into prime factors, we get

$Prime Factors$

180 = 2 × 2 × 3 × 3 × 5

(Here 2 and 3 are grouped in pairs of equal factors and 5 is not grouped in pairs of equal factors)

Therefore, 180 is not a perfect square.

2. Is 36 a perfect square? If so, find the number whose square is 36.

Solution:

Resolving 36 into prime factors, we get

$Prime Factors$

36 = 2 × 2 × 3 × 3.

Thus, 36 can be expressed as a product of pairs of equal factors.

Therefore, 36 is a perfect square.

Also, 36 = (2 × 3) × (2 × 3) = (6 × 6) = 6²

Hence, 6 is the number whose square is 36.

3. Is 196 a perfect square? If so, find the number whose square is 196.

Solution:

Resolving 196 into prime factors, we get

$Prime Factors$

196 = 2 x 2 x 7 x 7.

Thus, 196 can be expressed as a product of pairs of equal factors.

Therefore, 196 is a perfect square.

Also, 196 = (2 x 7) x (2 x 7) = (14 x 14) = (14)².

Hence, 14 is the number whose square is 196.

4. Show that 200 is not a perfect square.

Solution:

Resolving 200 into prime factors, we get

200 =2 x 2 x 2 x 5 x 5.

Making pairs of equal factors, we find that 2 is left.

Hence, 200 is not a perfect square.

5. Find the smallest number by which 252 must be multiplied to make it a perfect square.

Solution:

252 = 2 × 2 × 3 × 3 × 7

We observe that 2 and 3 are grouped in pairs and 7 is left unpaired.

If we multiply 252 by the factor 7 then,

252 × 7 = 2 × 2 × 3 × 3 × 7 × 7

1764 = 2 × 2 × 3 × 3 × 7 × 7, which is a perfect square.

Therefore, the required smallest number is 7.

6. Find the smallest number by which 396 must be divided so as to get a perfect square.

Solution:

396 = 2 × 2 × 3 × 3 × 11

We observe that 2 and 3 are grouped in pairs and 11 is left unpaired.

If we divide 396 by the factor 11 then,

396 ÷ 11 = (2 × 2 × 3 × 3 × 1̶1̶)/1̶1̶

= 2 × 2 × 3 × 3 = 36, which is a perfect square.

Therefore, the required smallest number is 11.