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) $16562=2\times 7\times 7\times 13\times 13$


Grouping the factors in pairs, we see that 2 is ;left unpaired
$\therefore$ Dividing by 2, we got the perfect square and sqaure root of the quotient will be $7\times 13=91$

(ii) $3698=2\times 43\times 43$

Grouping the factors in pairs, we see that 2 is left unpaired,
$\therefore$ Dividing 3698 by 2, the quotient will be = 43

(iii) $5103=3\times 3\times 3\times 3\times 3\times 3\times 7$

Grouping the factors in pairs, we see that 7 is left unpaired,
$\therefore$Dividing 5103 by 7, we get the quotient a perfect square.
and square root of the quotient will be $3\times 3\times 3=27$

(iv) $3174=2\times 3\times 23\times 23$

Grouping the factors in pairs, we see that 2 and 3 are left unpaired
$\therefore$ Dividing 3174 by $2\times 3$ i.e. 6, the quotient will be a perfect square and square root of the quotient will be = 23

(v) $1575=3\times 3\times 5\times 5\times 7$

Grouping the factors in pairs, we find that 7 is left unpaired
$\therefore$ dividing 1575 by 7, the quotient is a perfect square
and square root of the quotient will be $=3\times 5\times =15$