Substitute each of the capital letters by a different base six digit from 0 to 5 to satisfy the following set of alphametic relationships. None of the numbers can contain any leading zero.

- The base six number
**ICUS** is a perfect square, and:
**A** is a perfect square, and:
- The base six number
**MI** is a perfect fifth power, and:
- The base six number
**US** is a perfect square.

Determine the value assumed by the base six number

**AMICUS**.

MI must be 32 (base 10) = 52 (base 6)

A must be 1 or 4

US must be 25 (base 10) = 41 (base 6)

or 16 (base 10) = 24 (base 6)

or 9 (base 10) = 13 (base 6)

But US (base 6) cannot be 41 (because A is 1 or 4)

and cannot be 24 (because MI = 52)

so US (base 6) = 13

So A = 4

So C = 0 (the only remaining digit)

AMICUS = 452013

And, as a check, 2013 (base 6) = 441 (base 10) = 21*21