What positive integer n has the property that n³ and n⁴ together contain every digit from 0 to 9 exactly once?

We're looking for n such that n³ has four digits and n⁴ has six digits.

10000^(1/3) ~= 21.5 and 100000^(1/4) ~= 17.8, so the n must be between 18 and 21, inclusive.

It can't be 21 since n³ and n⁴ would both end in 1.

It can't be 20 since n³ and n⁴ would both end in 0.

It can't be 19 since n⁴ would begin and end in 1.

Therefore n must be 18.

To check, 18³ = 5832 and 18⁴ = 104976.

Edited on May 4, 2021, 7:36 am

Posted by tomarken on 2021-05-04 07:33:26