Problem:
Find the sum of the digits in the answer to
94 nines 9999⋯99×94 fours 4444⋯44
where a string of 94 nines is multiplied by a string of 94 fours.
Answer Choices:
A. 846
B. 855
C. 945
D. 954
E. 1072
Solution:
Since 94 nines 9999⋯99=194 zeros 0000⋯00−1, we have
94 nines9999⋯99×94 fours4444⋯44=(194 zeros0000⋯00−1)×94 fours4444⋯44=(94 fours4444⋯4494 zeros0000⋯00−94 fours4444⋯44)
which is
4444⋯440000⋯00− 4444⋯4493 fours 4444⋯4 3 93 fives 5555⋯56
The sum of the digits of this answer is
93(4)+3+93(5)+6=93(4+5)+(3+6)=94(9)=846.
OR
Try smaller cases to observe a pattern:
9×499×44999×4449999×4444⋮94 nines9999…99×94 fours4444…44======36435644355644435556⋮93 fours4444…4 3 93 fives5555…56sum=9=18=27=36=9×1=9×2=9×3=9×4=9×94
The sum of the digits of this answer is
93(4)+3+93(5)+6=93(4+5)+(3+6)=94(9)=846.
Query. What is the sum of the digits when any 94-digit number is multiplied by 94 nines9999…99?
Answer: A.
The problems on this page are the property of the MAA's American Mathematics Competitions