2021 Fall AMC 10B Problem 3

Problem 3 of 25EasierAlgebra

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?

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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 .