The expression (2021)/(2020) - (2020)/(2021) is equal to the fraction (p)/(q) in which p and q are positive integers whose greatest common divisor is 1. What is p?

2021 AMC 10B Fall Problem 3 — Solution

2021 AMC 10B Fall Problem 3

Below is the professionally curated solution for Problem 3 of the 2021 AMC 10B Fall, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 10B Fall solutions, or check the answer key.

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Concepts:fraction difference of squares

Difficulty rating: 770

The expression $\dfrac{2021}{2020} - \dfrac{2020}{2021}$ is equal to the fraction $\frac{p}{q}$ in which $p$ and $q$ are positive integers whose greatest common divisor is $1.$ What is $p?$

$1$

$9$

$2020$

$2021$

$4041$

Solution:

We can rewrite this as $\dfrac{2021\cdot 2021}{2020\cdot 2021} -\dfrac{2020\cdot 2020}{2020\cdot 2021}$ $= \dfrac{2021^2-2020^2}{2020\cdot 2021}.$

This can be simplified to $\dfrac{(2021-2020)(2021+2020)}{2020\cdot 2021}$ $=\dfrac{4041}{2020\cdot 2021}.$

Since $4041$ is coprime with both $2020$ and $2021,$ we know $4041$ is the numerator.

Thus, the answer is E .

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