The product of integers 420 and k is a perfect cube. What is the smallest possible positive value of k?

420 = 2*2*3*5*7
k = 2*9*25*49
= 22,050

420 = 2^2 × 3 × 5 ×7
k
= 2 × 3^2 × 5^2 × 7^2
= 22,050

If the product of 420 and k makes a perfect cube, then we need to find the prime factorization of 420 and then see what is needed to make the product of that and another number a cube:

420 = 2² * 3 * 5 * 7

For the product of 420 and k to be a cube, the exponents must all be a multiple of 3.

So the smallest such number is:

k = 2 * 3² * 5² * 7²
k = 2 * 9 * 25 * 49
k = 22050

The product of 420 and k is then:

420 * 22050 = 9261000

Which is a perfect cube:

∛9261000 = 210 (which is 2 * 3 * 5 * 7)