Question

Find the least number by which
(5)9 (3)6 x (17) has to be
multiplied to obtain a perfect
square.
Select one:
a. 5
b. 17
c. 221
d. 85​

Answer 1

85 is the least number by which 5⁹ × 3⁶ × 17 has to be multiplied to obtain a perfect square.

Correct question : Find the least number by which 5⁹ × 3⁶ × 17 has to be multiplied to obtain a perfect square a. 5 b. 17 c. 221 d. 85

Given :

The number 5⁹ × 3⁶ × 17

To find :

The least number by which 5⁹ × 3⁶ × 17 has to be multiplied to obtain a perfect square

a. 5

b. 17

c. 221

d. 85

Concept :

Step : I Firstly express the given number as a product of prime factor by using prime factorisation

Step : II Make the pair of similar factors such that the both factors in each pair are equal.

Step : III Take one factor from each pair.

Step : IV . If no factor is left over in grouping (pairs) then the number is perfect square otherwise not

Solution :

Step 1 of 2 :

Write down the given number

Here the given number is 5⁹ × 3⁶ × 17

Step 2 of 2 :

Find the least number to be multiplied to obtain a perfect square.

$\displaystyle \sf {5}^{9} \times {3}^{6} \times 17$

$\displaystyle \sf = {5}^{2} \times {5}^{2} \times {5}^{2} \times {5}^{2} \times {5}^{1} \times {3}^{2} \times {3}^{2} \times {3}^{2} \times {17}^{1}$

We see that in the above factorisation 5 and 17 are left over in grouping (pairs)

We need to multiply the given number by 5 × 17 i.e 85

Hence 85 is the least number by which 5⁹ × 3⁶ × 17 has to be multiplied to obtain a perfect square.

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