Question

Find the smallest number by which the given number must be divided so that the resulting number is a perfect square: 

(i) 12283 

(ii) 1800 

(iii) 2904

Answer 1

(i) 12283 

Resolving 14283 into prime factors, we get 

14283 = 3 × 3 × 3 × 23 × 23 

Obtained factors can be paired into equal factors except for 3 

So, eliminate 3 by diving the dividing the number with 3

\(\frac{14283}{3}\) = (3 × 3) × (23 × 23)

Again,

\(\frac{14283}{3}\) = (3 × 23) × (3 × 23)

= 69 × 69 

= (69)² 

Therefore, 

The resultant is the square of 69 

(ii) 1800 

Resolving 1800 into prime factors, we get 

1800 = 2 × 2 × 5 × 5 × 3 × 3 × 2 

Obtained factors can be paired into equal factors except for 2 

So, eliminate 2 by diving the dividing the number with 2

\(\frac{1800}{2}\) = (2 × 2) × (3 × 3) × (5 × 5)

Again, 

\(\frac{1800}{2}\)= (2 × 3 × 5) × (2 × 3 × 5) 

= 30 × 30 

= (30)² 

Therefore, The resultant is the square of 30.

(iii) 2904 

Resolving 2904 into prime factors, we get 

2904 = 2 × 2 × 11 × 11 × 2 × 3 

Obtained factors can be paired into equal factors except for 2 and 3 

So, eliminate 6 by diving the dividing the number with 6

\(\frac{2904}{6}\) = (2 × 2) × (11 × 11)

Again, 

\(\frac{2904}{6}\)= (2 × 11) × (2 × 11) 

= 22 × 22 

= (22)² 

Therefore, The resultant is the square of 22.

Answer 2

(i) 12283

Resolving 14283 into prime factors, we get

14283 = 3 × 3 × 3 × 23 × 23

Obtained factors can be paired into equal factors except for 3

So, eliminate 3 by diving the dividing the number with 3

 = (3 × 3) × (23 × 23)

Again,

 = (3 × 23) × (3 × 23)

= 69 × 69

= (69)²

Therefore,

The resultant is the square of 69

(ii) 1800

Resolving 1800 into prime factors, we get

1800 = 2 × 2 × 5 × 5 × 3 × 3 × 2

Obtained factors can be paired into equal factors except for 2

So, eliminate 2 by diving the dividing the number with 2

 = (2 × 2) × (3 × 3) × (5 × 5)

Again,

 = (2 × 3 × 5) × (2 × 3 × 5)

= 30 × 30

= (30)²

Therefore,

The resultant is the square of 30

(iii) 2904

Resolving 2904 into prime factors, we get

2904 = 2 × 2 × 11 × 11 × 2 × 3

Obtained factors can be paired into equal factors except for 2 and 3

So, eliminate 6 by diving the dividing the number with 6

 = (2 × 2) × (11 × 11)

Again,

 = (2 × 11) × (2 × 11)

= 22 × 22

= (22)²

Therefore,

The resultant is the square of 22