Problem 4

When

4444^{4444}

is written in decimal notation, the sum of its digits is

A

. Let

B

be the sum of the digits of

A

. Find the sum of the digits of

B

. (

A

and

B

are written in decimal notation.)

Step 4 of 6: Digit sums preserve residues modulo 9

Previous step Next step

N \equiv S(N) \pmod 9 \implies 4444^{4444} \equiv A \equiv B \equiv S(B) \pmod 9

Detailed analysis

Every power of $10$ is $\equiv 1 \pmod 9$ , so a number and its digit sum are always congruent modulo $9$ : writing $N = \sum_k a_k 10^k$ , we get $N \equiv \sum_k a_k = S(N) \pmod 9$ . Applying this three times, $4444^{4444} \equiv A \equiv B \equiv S(B) \pmod 9$ .