Problem:
The 7-digit numbers 7 4 A 5 2 B 1 and 3 2 6 A B 4 C are each multiples of 3. Which of the following could be the value of C?
Answer Choices:
A. 1
B. 2
C. 3
D. 5
E. 8
Solution:
For 7 4 A 5 2 B 1 to be a multiple of 3, the sum 7+4+A+5+ 2+B+1=19+A+B must be a multiple of 3. Therefore A+B is 1 less than a multiple of 3. The sum 3+2+6+A+B+4+C=15+A+B+C must be a multiple of 3, so A+B+C must be a multiple of 3. Since A+B is 1 less than a multiple of 3,C must be 1 more than a multiple of 3. Only choice (A) meets this requirement.
Answer: A.
The problems on this page are the property of the MAA's American Mathematics Competitions