find the number whose square is the new number. (i) 16562 (ii) 3698 (iii) 5103 (iv) 3174 (v) 1575

Question

By What numbers should each of the following be divided to get a perfect square in each case? Also, find the number whose square is the new number. 

(i) 16562 

(ii) 3698 

(iii) 5103 

(iv) 3174 

(v) 1575

Answer 1

(i) 16562 

16562 = (7 × 7) × (13 × 13) × 2 

\(\frac{16562}{2}\)= (7 × 7) × (13 × 13) 

\(\frac{16562}{2}\)= (7 × 13) × (7 × 13) = 91 × 91 

= 91² 

Therefore, the resultant is the square of 91.

ii) 3698 

3698 = 2 × (43 × 43) 

\(\frac{3698}{2}\)= 43 × 43 

= 43² 

Therefore, the numbers must be divided by 2 and resultant is square of 43.

(iii) 5103 

5103 = (3 × 3) × (3 × 3) × (3 × 3) × 7 

\(\frac{5103}{7} \)= (3 × 3 × 3) × (3 × 3 × 3) 

= 27 × 27 

= 27² 

Therefore, the number must be divided by 7 and resultant is square of 27.

(iv) 3174 

3174 = 2 × 3 × (23 × 23) 

\(\frac{3174}{6}\)= 23 × 23 

= 23² 

Therefore, the number must be divided by 6 and the resultant is square of 23.

(v) 1575

1575 = 3 × 3 × 5 × 5 × 7

\(\frac{1575}{7} \) = 3 × 3 × 5 × 5

= 15 × 15 

= 15² 

Therefore, the number must be divided by 7 and the resultant is square of 15.