2021 AMC 10 B Fall

Complete problem set with solutions and individual problem pages

Problem 3 Easy

The expression \frac{2021}{2020}-\frac{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 ?(2021 AMC Fall 10B, Question #3)

  • A.

    1

  • B.

    9

  • C.

    2020

  • D.

    2021

  • E.

    4041

Answer:E

Solution 1:

We write the given expression as a single fraction: \frac{2021}{2020}-\frac{2020}{2021}=\frac{2021 \cdot 2021-2020 \cdot 2020}{2020 \cdot 2021} by cross multiplication. Then by factoring the numerator, we get \frac{2021 \cdot 2021-2020 \cdot 2020}{2020 \cdot 2021}=\frac{(2021-2020)(2021+2020)}{2020 \cdot 2021} . The question is asking for the numerator, so our answer is 2021+2020=4041, giving answer choice (E).

Solution 2:

Denote a=2020. Hence, \begin{aligned} \frac{2021}{2020}-\frac{2020}{2021} &=\frac{a+1}{a}-\frac{a}{a+1} \\ &=\frac{(a+1)^{2}-a^{2}}{a(a+1)} \\ &=\frac{2 a+1}{a(a+1)} \end{aligned} We observe that \text{gcd}(2 a+1, a)=1 and \text{gcd}(2 a+1, a+1)=1. Hence, \text{gcd}(2 a+1, a(a+1))=1. Therefore, p=2 a+1=4041. Therefore, the answer is (E) 4041

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