By what number should each of the following numbers be multiplied to get a perfect square in each case? Also, find the number whose square is the new number.
(i) 8820
(ii) 3675
(iii) 605
(iv) 2880
(v) 4056
(vi) 3468

To do :

We have to find the numbers by which the given numbers must be multiplied 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.

(i) $8820=2\times2\times3\times3\times5\times7\times7$

$=(2)^2\times(3)^2\times5\times(7)^2$

$8820\times5=(2)^2\times(3)^2\times5\times(7)^2\times5$

$=(2\times3\times5\times7)^2$

$=(210)^2$

In order to make the pairs an even number of pairs, we have to multiply 8820 by 5, then the product will be the perfect square.

Therefore, 5 is the smallest number by which 8820 must be multiplied so that the product is a perfect square and the number whose square is the new number is 210.

(ii) $3675=3\times5\times5\times7\times7$

$=3\times(5)^2\times(7)^2$

$3675\times3=3\times(5)^2\times(7)^2\times3$

$=(3\times5\times7)^2$

$=(105)^2$

In order to make the pairs an even number of pairs, we have to multiply 3675 by 3, then the product will be the perfect square.

Therefore, 3 is the smallest number by which 3675 must be multiplied so that the product is a perfect square and the number whose square is the new number is 105. 

(iii) $605=5\times11\times11$

$=5\times(11)^2$

$605\times5=5\times(11)^2\times5$

$=(5\times11)^2$

$=(55)^2$

In order to make the pairs an even number of pairs, we have to multiply 605 by 5, then the product will be the perfect square.

Therefore, 5 is the smallest number by which 605 must be multiplied so that the product is a perfect square and the number whose square is the new number is 55. 

(iv) $2880=2\times2\times2\times2\times2\times2\times3\times3\times5$

$=(2)^2\times(2)^2\times(2)^2\times(3)^2\times5$

$2880\times5=(2)^2\times(2)^2\times(2)^2\times(3)^2\times(5)^2$

$=(2\times2\times2\times3\times5)^2$

$=(120)^2$

In order to make the pairs an even number of pairs, we have to multiply 2880 by 5, then the product will be the perfect square.

Therefore, 5 is the smallest number by which 2880 must be multiplied so that the product is a perfect square and the number whose square is the new number is 120. 

(v) $4056=2\times2\times2\times3\times13\times13$

$=(2)^2\times2\times3\times(13)^2$

$4056\times2\times3=(2)^2\times2\times3\times(13)^2\times2\times3$

$=(2\times2\times3\times13)^2$

$=(156)^2$

In order to make the pairs an even number of pairs, we have to multiply 4056 by 6, then the product will be the perfect square.

Therefore, 6 is the smallest number by which 4056 must be multiplied so that the product is a perfect square and the number whose square is the new number is 156. 

(vi) $3468=2\times2\times3\times17\times17$

$=(2)^2\times3\times(17)^2$

$3468\times3=(2)^2\times3\times(17)^2\times3$

$=(2\times3\times17)^2$

$=(102)^2$

In order to make the pairs an even number of pairs, we have to multiply 3468 by 3, then the product will be the perfect square.

Therefore, 3 is the smallest number by which 3468 must be multiplied so that the product is a perfect square and the number whose square is the new number is 102. 

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Updated on: 10-Oct-2022

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