Question

1. find the smallest numbers by which each of the following numbers must be multiplied to obtain a perfect cube:

(i) 121000 (ii) 43200 (iii) 5324 (iv) 1323

1. find the smallest numbers by which each of the following numbers must be divided to obtain a perfect cube.

(i) 1715 (ii) 3993 (iii) 3584 (iv) 17496

Explanation:

5.

(i) For 121000:

Step1: Prime - factorize the number

$121000 = 2^{3}\times5^{3}\times11^{2}$

Step2: Determine the missing factor

To make it a perfect cube, since the exponent of 11 is 2, we need one more 11. So the number to be multiplied is 11.

(ii) For 43200:

Step1: Prime - factorize the number

$43200=2^{7}\times3^{3}\times5^{2}$

Step2: Determine the missing factor

For the exponent of 2, $7 = 3\times2 + 1$, for the exponent of 5, $2$ is the exponent. We need $2^{2}\times5^{1}= 20$ to make the exponents of all prime - factors multiples of 3.

(iii) For 5324:

Step1: Prime - factorize the number

$5324 = 2^{2}\times11^{3}$

Step2: Determine the missing factor

Since the exponent of 2 is 2, we need one more 2. So the number to be multiplied is 2.

(iv) For 1323:

Answer:

(i) 11
(ii) 20
(iii) 2
(iv) 7

6.

(i) For 1715: