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 3 of 6: Bound

S(B)

by maximizing the digit sum over

B \le 45

Previous step Next step

B \le 45 \implies S(B) \le S(39) = 12

Detailed analysis

Among the integers from $1$ to $45$ , the largest digit sum belongs to $39$ (since a two-digit number $\overline{tu} \le 45$ has $t \le 3$ except when $t=4$ forces $u \le 5$ , and $3+9=12 > 4+5=9$ ), giving $S(39) = 12$ . Hence $S(B) \le 12$ .