Problem: In base R1​ the expanded fraction F1​ becomes .373737⋯ and the expanded fraction F2​ becomes .737373⋯. In base R2​ fraction F1​, when expanded, becomes .252525⋯ while fraction F2​ becomes .525252⋯. The sum of R1​ and R2​, each written in the base ten, is:
Answer Choices:
A. 24
B. 22
C. 21
D. 20
E. 19
Solution:
F1​=R12​−13R1​+7​=R22​−12R2​+5​ and F2​=R12​−17R1​+3​=R22​−15R2​+2​∴2R2​+53R1​+7​=R22​−1R12​−1​=5R2​+27R1​+3​
∴R2​=R1​+2929R1​+1​ Knowing that R1​ and R2​ must each be integral, and that R1​⩾8 (why?), we solve for R2​ with permissible values of R1​. For R1​=8,R2​ is not integral; for R1​=9 or 10,R2​ is not integral; for R1​=11,R2​=8; for R1​=12,R2​ is not integral; for R1​=13,R2​=9. The values R1​=13,R2​=9 do not satisfy the conditions of the problem; the values R1​=11,R2​=8 do. ∴R1​+R2​=19