Question

the smallest number which when multiplied with 7200 will make the product a perfect cube is

Answer 1

30 is the smallest no.

Answer 2

The smallest number that when multiplied by 7200 makes it a perfect cube is 30.

Given:

The number 7200.

To Find:

The smallest number that when multiplied by 7200 makes it a perfect cube.

Solution:

A perfect cube is a number that can be expressed as powers of 3 and its multiples. So we need to find a number that when multiplied by 7200 gives the resultant number that can be expressed as a power of multiples of 3.

On factorizing 7200 we get

7200 =  2 x 2 x 2 x 2 x 2 x 3 x 3 x 5 x 5 = $2^{5}$ x 3² x 5²

Each of the prime factor should have power as multiple of 3.

$2^{5}$  x 2 = $2^{6}$

3² x 3 = 3³

5² x 5 = 5³

On multiplying  (2 x 3 x 5) to 7200 we get

7200 x  (2 x 3 x 5) = $2^{6}$ x 3³x  5³ = 60³

Hence, we need to multiply 30 to 7200 make it a perfect cube.

∴The smallest number that when multiplied by 7200 makes it a perfect cube is 30.

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