Question

Using prime factorization method, find which of the following numbers are perfect squares?

189, 225, 2048, 343, 441, 2961, 11025, 3549

Answer 1

Since,

189 = 3² × 3 × 7 

It cannot be written as pair of two equal factors, so

189 is not a perfect square.

Since,

225 = (5 × 5) × (3 × 3) 

It can be written as pair of two equal factors, so

22 is a perfect square.

Since, 

2048 = (2 × 2) × (2 × 2) × (2 × 2) (2 × 2) × (2 × 2) × 2

 All the factors cannot be written as pair of two equal factors, so

189 is not a perfect square.

Since, 

343 = (7 × 7) × 7 

It cannot be written as pair of two equal factors, so

343 is not a perfect square.

Since, 

441 = (7 × 7) × (3 × 3) 

It can be written as pair of two equal factors, so 

441 is a perfect square. 

Since, 

2916 = (3 × 3) × (3 × 3) × (3 × 3) × (2 × 2) 

It can be written as pair of two equal factors, so 

2916 is a perfect square.

Since, 

11025 = (5 × 5) × (3 × 3) × (7 × 7) 

It can be written as pair of two equal factors, so

 11025 is a perfect square.

Since, 

3549 = (13 × 13) × 3 × 7 

It cannot be written as pair of two equal factors, so 

3549 is not a perfect square.