Difference between revisions of "2020 AMC 10B Problems/Problem 22"

(MAA Official Solution)
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<math>\textbf{(A) } 100 \qquad\textbf{(B) } 101 \qquad\textbf{(C) } 200 \qquad\textbf{(D) } 201 \qquad\textbf{(E) } 202</math>
 
<math>\textbf{(A) } 100 \qquad\textbf{(B) } 101 \qquad\textbf{(C) } 200 \qquad\textbf{(D) } 201 \qquad\textbf{(E) } 202</math>
  
==Solution==
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==Solution 1==
  
 
Let <math>x=2^{50}</math>. We are now looking for the remainder of <math>\frac{4x^4+202}{2x^2+2x+1}</math>.
 
Let <math>x=2^{50}</math>. We are now looking for the remainder of <math>\frac{4x^4+202}{2x^2+2x+1}</math>.
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Since the first half divides cleanly as shown earlier, the remainder must be <math>\boxed{\textbf{(D) }201}</math> ~quacker88
 
Since the first half divides cleanly as shown earlier, the remainder must be <math>\boxed{\textbf{(D) }201}</math> ~quacker88
  
==Solution 2==
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==Solution 2 (MAA Original Solution)==
Similar to Solution 1, let <math>x=2^{50}</math>. It suffices to find remainder of <math>\frac{4x^4+202}{2x^2+2x+1}</math>. Dividing polynomials results in a remainder of <math>\boxed{\textbf{(D) } 201}</math>.
 
  
==MAA Official Solution==
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<cmath>
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\begin{align*}
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2^{202} + 202 &= (2^{101})^2 + 2\cdot 2^{101} + 1 - 2\cdot 2^{101} + 201\\
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&= (2^{101} + 1)^2 - 2^{102} + 201\\
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&= (2^{101} - 2^{51} + 1)(2^{101} + 2^{51} + 1) + 201.
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\end{align*}
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</cmath>
  
<cmath>2^{202} + 202 = (2^{101})^2 + 2\cdot 2^{101} + 1 - 2\cdot 2^{101} + 201</cmath>
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Thus, we see that the remainder is surely <math>\boxed{\textbf{(D) } 201}</math>
                          <cmath>= (2^{101} + 1)^2 - 2^{102} + 201</cmath>
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                          <cmath>= (2^{101} - 2^{51} + 1)(2^{101} + 2^{51} + 1) + 201.</cmath>
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(Source: https://artofproblemsolving.com/community/c5h2001950p14000817)
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==Solution 3 ==
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We let <cmath>x = 2^{50}</cmath> and <cmath>2^{202} + 202 = 4x^{4} + 202</cmath>.
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Next we write <cmath>2^{101} + 2^{51} + 1 = 2x^{2} + 2x + 1</cmath>.
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We know that <cmath>4x^{4} + 1 = (2x^{2} + 2x + 1)(2x^{2} - 2x + 1)</cmath> by the Sophie Germain identity so to find <cmath>4x^{4} + 202,</cmath> we find that <cmath>4x^{4} + 202 = 4x^{4} + 201 + 1</cmath> which shows that the remainder is <math>\boxed{\textbf{(D) } 201}</math>
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==Solution 4 ==
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We let <math>x=2^{50.5}</math>. That means <math>2^{202}+202=x^{4}+202</math> and <math>2^{101}+2^{51}+1=x^{2}+x\sqrt{2}+1</math>. Then, we simply do polynomial division, and find that the remainder is <math>\boxed{\textbf{(D) } 201}</math>.
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==Solution 5 (Modular Arithmetic)==
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Let <math>n=2^{101}+2^{51}+1</math>. Then,
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<math>2^{202}+202 \equiv (-2^{51}-1)^2 + 202 \pmod{n}</math>
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<math>\equiv 2^{102}+2^{52}+203 \pmod{n}</math>
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<math>= 2(n-1)+203 \equiv 201 \pmod{n}</math>.  
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Thus, the remainder is <math>\boxed{\textbf{(D) } 201}</math>.
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~ Leo.Euler
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 +
~ (edited by asops)
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==Video Solutions==
 +
 
 +
===Video Solution 1 by Mathematical Dexterity (2 min)===
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https://www.youtube.com/watch?v=lLWURnmpPQA
 +
 
 +
===Video Solution 2 by The Beauty Of Math===
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https://youtu.be/gPqd-yKQdFg
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 +
===Video Solution 3===
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https://www.youtube.com/watch?v=Qs6UnryIAI8&list=PLLCzevlMcsWNcTZEaxHe8VaccrhubDOlQ&index=9&t=0s ~ MathEx
 +
 
 +
===Video Solution 4 Using Sophie Germain's Identity===
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https://youtu.be/ba6w1OhXqOQ?t=5155
 +
 
 +
~ pi_is_3.14
  
 
==See Also==  
 
==See Also==  

Revision as of 20:43, 31 October 2021

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

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 denominator 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 2 (MAA Original Solution)

\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 surely $\boxed{\textbf{(D) } 201}$

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

Solution 3

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 $2^{101}+2^{51}+1=x^{2}+x\sqrt{2}+1$. Then, we simply do polynomial division, and find that the remainder is $\boxed{\textbf{(D) } 201}$.

Solution 5 (Modular Arithmetic)

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

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

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

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

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

~ Leo.Euler

~ (edited by asops)

Video Solutions

Video Solution 1 by Mathematical Dexterity (2 min)

https://www.youtube.com/watch?v=lLWURnmpPQA

Video Solution 2 by The Beauty Of Math

https://youtu.be/gPqd-yKQdFg

Video Solution 3

https://www.youtube.com/watch?v=Qs6UnryIAI8&list=PLLCzevlMcsWNcTZEaxHe8VaccrhubDOlQ&index=9&t=0s ~ MathEx

Video Solution 4 Using Sophie Germain's Identity

https://youtu.be/ba6w1OhXqOQ?t=5155

~ pi_is_3.14

See Also

2020 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 21
Followed by
Problem 23
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions

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