1994 AMC 8 Problem 25

Question

Find the sum of the digits in the answer to

\underbrace{9999\cdots99}_{94} \times \underbrace{4444\cdots44}_{94}

where a string of

94

nines is multiplied by a string of

94

fours.

846

855

945

954

1072

Try small cases first:

9 \times 4,

99 \times 44,

999 \times 444

The product is a block of

4

s, then a

3,

then a block of

5

s, then a

6

; count how many of each

Small cases show the pattern:

99 \times 44 = 4356

and

999 \times 444 = 443556.

In general, a string of

n

nines times a string of

n

fours gives

(n-1)

fours, then a

3,

then

(n-1)

fives, then a

6.

For

n = 94,

the product has

93

fours, one

3,

93

fives, and one

6.

The digit sum is

93(4) + 3

+ 93(5) + 6

= 93(9) + 9

= 94(9)

= 846.

Thus, the correct answer is A .

Problem 25 in other years