To find the smallest number by which 2592 should be multiplied so that the product is a perfect cube, we need to factorize 2592 into its prime factors and then find the factors that will make it a perfect cube.
2592 can be factorized as follows:
2592 = 2^5 * 3^4
To make it a perfect cube, we need to ensure that each prime factor appears in multiples of three. So, we need to add 2 more powers of 2 and 2 more powers of 3 to make the exponents multiples of 3:
2^5 * 3^4 * 2^1 * 3^2 = 2^6 * 3^6
So, the smallest number by which 2592 should be multiplied to make it a perfect cube is 2^1 x 3^2 = 2 x 9 = 18.
