Problem:
There exist positive integers A,B and C, with no common factor greater than 1, such that
Alog2005+Blog2002=C
What is A+B+C?
Answer Choices:
A. 6
B. 7
C. 8
D. 9
E. 10
Solution:
Note that
C=Alog2005+Blog2002=log2005A+log2002B=log200(5A⋅2B),
so 200C=5A⋅2B. Therefore, 5A⋅2B=200C=(52⋅23)C=52C23C. By uniqueness of prime factorization, †A=2C and B=3C. Letting C=1 we get A=2,B=3 and A+B+C=6. The triplet (A,B,C)=(2,3,1) is the only solution with no common factor greater than 1.