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 2 of 6: Bound

B

by maximizing the digit sum over

A\le 159984

Previous step Next step

A \le 159984 \implies B=S(A) \le S(99999)=45

Detailed analysis

Among all integers from $1$ to $159984$ , the one with the largest digit sum is $99999$ (five nines), since a $6$ -digit number in this range must start with $1$ and adding more nines to a shorter number can never beat five nines: $S(99999) = 45$ . Hence $B = S(A) \le 45$ .