Browse · MathNet
PrintFall 2021 AMC 10 B
United States 2021 algebra
Problem
The expression is equal to the fraction in which and are positive integers whose greatest common divisor is 1. What is ? (A) 1 (B) 9 (C) 2020 (D) 2021 (E) 4041
Solution
The given expression equals
Because $4041 - 2 \cdot 2020 = 1$, it follows that $4041$ and $2020$ cannot have a common divisor greater than $1$. Similarly, because $4041 - 2 \cdot 2021 = -1$, it follows that $4041$ and $2021$ cannot have a common divisor greater than $1$. Hence $\frac{4041}{2020 \cdot 2021}$ is in simplest terms, and the requested numerator is $4041$. Let $n = 2020$. Then the given fraction equals $daa+bdbnn+112n+11nn+1p = 2n + 1 = 4041$.
Because $4041 - 2 \cdot 2020 = 1$, it follows that $4041$ and $2020$ cannot have a common divisor greater than $1$. Similarly, because $4041 - 2 \cdot 2021 = -1$, it follows that $4041$ and $2021$ cannot have a common divisor greater than $1$. Hence $\frac{4041}{2020 \cdot 2021}$ is in simplest terms, and the requested numerator is $4041$. Let $n = 2020$. Then the given fraction equals $daa+bdbnn+112n+11nn+1p = 2n + 1 = 4041$.
Final answer
E
Techniques
FractionsGreatest common divisors (gcd)Polynomial operations