2020 AMC 10B Problem 22

Question

What is the remainder when

2^{202} +202

is divided by

2^{101}+2^{51}+1?

100

101

200

201

202

Try to factor the numerator using a difference of squares

Write

2^{202}+202

as a multiple of the divisor plus a small remainder

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Let

m=2^{101}+2^{51}+1.

We factor the numerator around this divisor:

\begin{aligned} &2^{202}+202 \\ &\quad =(2^{101}+1)^2-(2^{51})^2+201. \end{aligned}

By the difference of squares,

\begin{aligned} &(2^{101}+1)^2-(2^{51})^2 \\ &\quad =(2^{101}+2^{51}+1) \\ &\quad {}\cdot(2^{101}-2^{51}+1), \end{aligned}

which is a multiple of

m.

Therefore

2^{202}+202\equiv 201\pmod m.

Thus, the correct answer is D .

Problem 22 in other years