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Difference between revisions of "2021 Fall AMC 12A Problems/Problem 10"

(→‎Solution 2 (9's Identity))
m (You don't have to do that here, you can just add all the numbers to get 3.)
 
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<math>\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) }4</math>
 
<math>\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) }4</math>
  
−
==Solution 1==
+
==Solution 1 (Modular Arithmetic)==
 
Recall that <math>9\equiv-1\pmod{5}.</math> We expand <math>N</math> by the definition of bases:
 
Recall that <math>9\equiv-1\pmod{5}.</math> We expand <math>N</math> by the definition of bases:
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
 
N&=27{,}006{,}000{,}052_9 \\
 
N&=27{,}006{,}000{,}052_9 \\
 
&= 2\cdot9^{10} + 7\cdot9^9 + 6\cdot9^6 + 5\cdot9 + 2 \\
 
&= 2\cdot9^{10} + 7\cdot9^9 + 6\cdot9^6 + 5\cdot9 + 2 \\
−
&\equiv 2\cdot(-1)^{10} + 7\cdot(-1)^9 + 6\cdot(-1)^6 + 5\cdot(-1) + 2 &&\pmod{5} \\
+
&\equiv 2\cdot(-1)^{10} + 7\cdot(-1)^9 + 6\cdot(-1)^6 + 2 &&\pmod{5} \\
−
&= 2-7+6-5+2 \\
+
&\equiv 2-7+6+2 &&\pmod{5} \\
−
&= -2 \\
 
 
&\equiv \boxed{\textbf{(D) } 3} &&\pmod{5}.
 
&\equiv \boxed{\textbf{(D) } 3} &&\pmod{5}.
 
\end{align*}</cmath>
 
\end{align*}</cmath>
−
~Aidensharp ~kante314 ~MRENTHUSIASM
+
~Aidensharp ~Kante314 ~MRENTHUSIASM ~anabel.disher
  
−
==Solution 2 (9's Identity)==
+
==Solution 2 (Powers of 9)==
  
−
We need to first convert <math>N</math> into a regular base-10 integer:
+
We need to first convert <math>N</math> into a regular base-<math>10</math> number: <cmath>N = 27{,}006{,}000{,}052_9 = 2\cdot9^{10} + 7\cdot9^9 + 6\cdot9^6 + 5\cdot9 + 2.</cmath>
  
 +
Now, consider how the last digit of <math>9</math> changes with changes of the power of <math>9:</math>
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
−
N&=27{,}006{,}000{,}052_9 \\
+
9^0&=1, \\
−
&= 2\cdot9^{10} + 7\cdot9^9 + 6\cdot9^6 + 5\cdot9 + 2 \\
+
9^1&=9, \\
 +
9^2&=\ldots 1, \\
 +
9^3&=\ldots 9, \\
 +
9^4&=\ldots 1, \\
 +
& \ \vdots
 
\end{align*}</cmath>
 
\end{align*}</cmath>
 +
Note that if <math>x</math> is odd, then <math>9^x \equiv 4\pmod{5}.</math> On the other hand, if <math>x</math> is even, then <math>9^x \equiv 1\pmod{5}.</math>
  
−
Now, consider how the last digit of <math>9</math> changes with changes of the power of <math>9</math>:
+
Therefore, we have
−
 
 
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
−
9^0=1 \\
+
N&\equiv 2\cdot(1) + 7\cdot(4) + 6\cdot(1) + 5\cdot(4) + 2\cdot(1) &&\pmod{5} \\
−
9^1=9 \\
+
&\equiv 2+28+6+20+2 &&\pmod{5} \\
−
9^2=81 \\
+
&\equiv 58 &&\pmod{5} \\
−
9^3=729 \\
+
&\equiv \boxed{\textbf{(D) } 3} &&\pmod{5}. \\
−
9^4=6561 \\
 
−
......
 
 
\end{align*}</cmath>
 
\end{align*}</cmath>
 +
Note that for the odd case, <math>9^x \equiv -1\pmod{5}</math> may simplify the process further, as given by Solution 1.
 +
 +
~Wilhelm Z
 +
 +
==Remark==
 +
By the long division algorithm, you can work from left to right accumulating the answer, and don't need to count the digits to get the power. You only need to take the current leading digit modulo <math>5</math>, negate it, and add it to the next digit, and repeat the process until you reach the units digit.
  
−
Note that if <math>x</math> is odd:
+
~oinava
  
−
<cmath>\begin{align*}
+
==Video Solution ==
−
9^x &\equiv 4\pmod{5} \\
+
https://youtu.be/Mv38a8oanFk
−
\end{align*}</cmath>
 
  
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If <math>x</math> is even:
+
~Education, the Study of Everything
  
−
<cmath>\begin{align*}
+
== Video Solution==
−
9^x &\equiv 1\pmod{5} \\
+
https://youtu.be/SCGzEOOICr4?t=101
−
\end{align*}</cmath>
 
  
−
Therefore, we have:
+
==Video Solution ==
 +
https://youtu.be/zq3UPu4nwsE?t=358
  
−
<cmath>\begin{align*}
+
https://youtu.be/wlDlByKI7A8?t=885
−
N&\equiv 2\cdot(1) + 7\cdot(4) + 6\cdot(1) + 5\cdot(4) + 2\cdot(1) &\pmod{5} \\
 
−
&\equiv 2+28+6+20+2 &\pmod{5} \\
 
−
&\equiv 58 &\pmod{5} \\
 
−
&\equiv \boxed{\textbf{(D) } 3} &\pmod{5} \\
 
−
\end{align*}</cmath>
 
  
−
Note that for the odd case, <math>9^x \equiv -1\pmod{5}</math> may simplify the process further, as given by Solution 1.
+
==Video Solution by WhyMath==
 +
https://youtu.be/4faUhsTmS5A
  
−
~Wilhelm Z
+
~savannahsolver
  
 
==See Also==
 
==See Also==

Latest revision as of 14:38, 13 August 2024

The following problem is from both the 2021 Fall AMC 10A #12 and 2021 Fall AMC 12A #10, so both problems redirect to this page.

Problem

The base-nine representation of the number $N$ is $27{,}006{,}000{,}052_{\text{nine}}.$ What is the remainder when $N$ is divided by $5?$

$\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) }4$

Solution 1 (Modular Arithmetic)

Recall that $9\equiv-1\pmod{5}.$ We expand $N$ by the definition of bases: \begin{align*} N&=27{,}006{,}000{,}052_9 \\ &= 2\cdot9^{10} + 7\cdot9^9 + 6\cdot9^6 + 5\cdot9 + 2 \\ &\equiv 2\cdot(-1)^{10} + 7\cdot(-1)^9 + 6\cdot(-1)^6 + 2 &&\pmod{5} \\ &\equiv 2-7+6+2 &&\pmod{5} \\ &\equiv \boxed{\textbf{(D) } 3} &&\pmod{5}. \end{align*} ~Aidensharp ~Kante314 ~MRENTHUSIASM ~anabel.disher

Solution 2 (Powers of 9)

We need to first convert $N$ into a regular base-$10$ number: \[N = 27{,}006{,}000{,}052_9 = 2\cdot9^{10} + 7\cdot9^9 + 6\cdot9^6 + 5\cdot9 + 2.\]

Now, consider how the last digit of $9$ changes with changes of the power of $9:$ \begin{align*} 9^0&=1, \\ 9^1&=9, \\ 9^2&=\ldots 1, \\ 9^3&=\ldots 9, \\ 9^4&=\ldots 1, \\ & \ \vdots \end{align*} Note that if $x$ is odd, then $9^x \equiv 4\pmod{5}.$ On the other hand, if $x$ is even, then $9^x \equiv 1\pmod{5}.$

Therefore, we have \begin{align*} N&\equiv 2\cdot(1) + 7\cdot(4) + 6\cdot(1) + 5\cdot(4) + 2\cdot(1) &&\pmod{5} \\ &\equiv 2+28+6+20+2 &&\pmod{5} \\ &\equiv 58 &&\pmod{5} \\ &\equiv \boxed{\textbf{(D) } 3} &&\pmod{5}. \\ \end{align*} Note that for the odd case, $9^x \equiv -1\pmod{5}$ may simplify the process further, as given by Solution 1.

~Wilhelm Z

Remark

By the long division algorithm, you can work from left to right accumulating the answer, and don't need to count the digits to get the power. You only need to take the current leading digit modulo $5$, negate it, and add it to the next digit, and repeat the process until you reach the units digit.

~oinava

Video Solution

https://youtu.be/Mv38a8oanFk

~Education, the Study of Everything

Video Solution

https://youtu.be/SCGzEOOICr4?t=101

Video Solution

https://youtu.be/zq3UPu4nwsE?t=358

https://youtu.be/wlDlByKI7A8?t=885

Video Solution by WhyMath

https://youtu.be/4faUhsTmS5A

~savannahsolver

See Also

2021 Fall AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
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 12 Problems and Solutions
2021 Fall AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
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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