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

To do :

We have to find the numbers by which each of the given numbers must be divided so that the products are perfect squares and the numbers whose squares are the new numbers.

Solution:

Perfect Square: A perfect square has each distinct prime factor occurring an even number of times.

$16562=2\times7\times7\times13\times13$

$=2\times(7)^2\times(13)^2$

$16562\div2=2\times(7)^2\times(13)^2\div2$

$=(7\times13)^2$

$=(91)^2$

In order to make the pairs an even number of pairs, we have to divide 16562 by 2, then the product will be the perfect square.

Therefore, 2 is the smallest number by which 16562 must be divided so that the product is a perfect square and the number whose square is the new number is 91.

(ii) $3698=2\times43\times43$

$=2\times(43)^2$

$3698\div2=2\times(43)^2\div2$

$=(43)^2$

In order to make the pairs an even number of pairs, we have to divide 3698 by 2, then the product will be the perfect square.

Therefore, 2 is the smallest number by which 3698 must be divided so that the product is a perfect square and the number whose square is the new number is 43.

(iii) $5103=3\times3\times3\times3\times3\times3\times7$

$=(3)^2\times(3)^2\times(3)^2\times7$

$5103\div7=(3)^2\times(3)^2\times(3)^2\times7\div7$

$=(3\times3\times3)^2$

$=(27)^2$

In order to make the pairs an even number of pairs, we have to divide 5103 by 7, then the product will be the perfect square.

Therefore, 7 is the smallest number by which 5103 must be divided so that the product is a perfect square and the number whose square is the new number is 27.

(iv) $3174=2\times3\times23\times23$

$=6\times(23)^2$

$3174\div6=(23)^2\times6\div6$

$=(23)^2$

In order to make the pairs an even number of pairs, we have to divide 3174 by 6, then the product will be the perfect square.

Therefore, 6 is the smallest number by which 3174 must be divided so that the product is a perfect square and the number whose square is the new number is 23.

(v) $1575=3\times3\times5\times5\times7$

$=(3)^2\times(5)^2\times7$

$1575\div7=(3)^2\times(5)^2\times7\div7$

$=(3\times5)^2$

$=(15)^2$

In order to make the pairs an even number of pairs, we have to divide 1575 by 7, then the product will be the perfect square.

Therefore, 7 is the smallest number by which 1575 must be divided so that the product is a perfect square and the number whose square is the new number is 15.

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Updated on: 10-Oct-2022

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