2020 AMC 10B Problems/Problem 22

Problem

What is the remainder when $2^{202} +202$ is divided by $2^{101}+2^{51}+1$ ?

$\textbf{(A) } 100 \qquad\textbf{(B) } 101 \qquad\textbf{(C) } 200 \qquad\textbf{(D) } 201 \qquad\textbf{(E) } 202$

Solution 1 (MAA Original Solution)

Completing the square, then difference of squares:

$\begin{align*} 2^{202} + 202 &= (2^{101})^2 + 2\cdot 2^{101} + 1 - 2\cdot 2^{101} + 201\\ &= (2^{101} + 1)^2 - 2^{102} + 201\\ &= (2^{101} - 2^{51} + 1)(2^{101} + 2^{51} + 1) + 201. \end{align*}$

Thus, we see that the remainder is $\boxed{\textbf{(D) } 201}$

(Source: https://artofproblemsolving.com/community/c5h2001950p14000817)

Solution 2

Let $x=2^{50}$ . We are now looking for the remainder of $\frac{4x^4+202}{2x^2+2x+1}$ .

We could proceed with polynomial division, but the numerator looks awfully similar to the Sophie Germain Identity, which states that $a^4+4b^4=(a^2+2b^2+2ab)(a^2+2b^2-2ab)$

Let's use the identity, with $a=1$ and $b=x$ , so we have

$1+4x^4=(1+2x^2+2x)(1+2x^2-2x)$

Rearranging, we can see that this is exactly what we need:

$\frac{4x^4+1}{2x^2+2x+1}=2x^2-2x+1$

So $\frac{4x^4+202}{2x^2+2x+1} = \frac{4x^4+1}{2x^2+2x+1} +\frac{201}{2x^2+2x+1}$

Since the first half divides cleanly as shown earlier, the remainder must be $\boxed{\textbf{(D) }201}$

~quacker88

Solution 3 (Same As Solution 2)

We let $x = 2^{50}$ and $2^{202} + 202 = 4x^{4} + 202$ . Next we write $2^{101} + 2^{51} + 1 = 2x^{2} + 2x + 1$ . We know that $4x^{4} + 1 = (2x^{2} + 2x + 1)(2x^{2} - 2x + 1)$ by the Sophie Germain identity so to find $4x^{4} + 202,$ we find that $4x^{4} + 202 = 4x^{4} + 201 + 1$ which shows that the remainder is $\boxed{\textbf{(D) } 201}$

Solution 4

We let $x=2^{50.5}$ . That means $2^{202}+202=x^{4}+202$ and $2^{101}+2^{51}+1=x^{2}+2^{0.5}x+1$ . Then, we simply do polynomial division, and find that the remainder is $\boxed{\textbf{(D) } 201}$ .

The long division is essentially the same if you work with $x=2$ , or do repeated multiplication and subtraction using the original expression.

Solution 5 (Modular Arithmetic)

Let $n=2^{101}+2^{51}+1$ . Then, mod $n$ :

$2^{202}+202 \equiv (-2^{51}-1)^2 + 202$

$\equiv 2^{102}+2^{52}+203$

$= 2(n-1)+203 \equiv 201 \pmod{n}$ .

Thus, the remainder is $\boxed{\textbf{(D) } 201}$ .

~ Leo.Euler

~ (edited by asops)

Solution 6 (by Shiva Kumar Kannan - Least insightful & very straightforward + Manipulation)

We can repeatedly manipulate the numerator to make parts of it divisible by the denominator:

$\frac{2^{202}+202}{2^{101}+2^{51}+1}$ $= \frac{2^{202} + 2^{152} + 2^{101}}{2^{101}+2^{51}+1} - \frac{2^{152} + 2^{101} - 202}{2^{101}+2^{51}+1}$ $= 2^{101} - \frac{2^{152} + 2^{101} - 202}{2^{101}+2^{51}+1}$ $=2^{101} - \frac{2^{152}+2^{101}+2^{101}+2^{51} - 2^{101} - 2^{51} - 202}{2^{101}+2^{51}+1}$ $=2^{101} - 2^{51} + \frac{2^{101}+2^{51}+202}{2^{101}+2^{51}+1}$ $= 2^{101} - 2^{51} + \frac{2^{101}+2^{51}+1+201}{2^{101}+2^{51}+1}$ $= 2^{101} - 2^{51} + 1 + \frac{201}{2^{101} + 2^{51} + 1}.$

Clearly, $201 < 2^{201} + 2^{51} + 1$ , hence, we cannot manipulate the numerator further to make the denominator divide into one of its parts. This concludes, that the remainder is $\boxed{\textbf{(D) } 201}$ .

Video Solutions

~ pi_is_3.14

2020 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 21	Followed by Problem 23
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2020 AMC 10B Problems/Problem 22

Contents

Problem

Solution 1 (MAA Original Solution)

Solution 2

Solution 3 (Same As Solution 2)

Solution 4

Solution 5 (Modular Arithmetic)

Solution 6 (by Shiva Kumar Kannan - Least insightful & very straightforward + Manipulation)

Video Solutions

Video Solution 1 by Mathematical Dexterity (2 min)

Video Solution 2 by The Beauty Of Math

Video Solution 3

Video Solution 4 Using Sophie Germain's Identity

See Also