Problem of the Week 1116

The Digits of 4444⁴⁴⁴⁴

The sum of the digits of 4444⁴⁴⁴⁴ is A (in base 10). The sum of the digits of A is B. What is the sum of the digits in B?

I know that we could ask our favorite CAS, but there is a relatively short pencil proof.

Source: Mathematical Morsels by Ross Honsberger, 1978.

© Copyright 2009 Dave Ehren and Shilad Sen. Reproduced with permission.