Difference between revisions of "2012 AMC 8 Problems/Problem 12"

Latest revision as of 18:20, 23 November 2024

Problem

What is the units digit of $13^{2012}$ ?

$\textbf{(A)}\hspace{.05in}1\qquad\textbf{(B)}\hspace{.05in}3\qquad\textbf{(C)}\hspace{.05in}5\qquad\textbf{(D)}\hspace{.05in}7\qquad\textbf{(E)}\hspace{.05in}9$

Video Solution by Pi Academy

https://youtu.be/7xlBdxcsP3I?si=sFJGKXnBUUN7Su-C

Video Solution 2

https://youtu.be/6RGNZj0tt2w ~David

https://youtu.be/7an5wU9Q5hk?t=1186 ~ pi_is_3.14

https://youtu.be/6c_s967T7cA ~savannahsolver

Solution 1

The problem wants us to find the units digit of $13^{2012}$ , therefore, we can eliminate the tens digit of $13$ , because the tens digit will not affect the final result. So our new expression is $3^{2012}$ . Now we need to look for a pattern in the units digit.

$3^1 \implies 3$

$3^2 \implies 9$

$3^3 \implies 7$

$3^4 \implies 1$

$3^5 \implies 3$

We observe that there is a pattern for the units digit which recurs every four powers of three. Using this pattern, we can subtract 1 from 2012 and divide by 4. The remainder is the power of three that we are looking for, plus one. $2011$ divided by $4$ leaves a remainder of $3$ , so the answer is the units digit of $3^{3+1}$ , or $3^4$ . Thus, we find that the units digit of $13^{2012}$ is $\boxed{{\textbf{(A)}\ 1}}$ .

Solution 2

Ignore the tens digit of $13$ , we find a pattern in the units digit that $3^4 \implies 1$ . We also find $2012$ can be divided by $4$ evenly, which is $2012/4=503$ . So $3^{2012}$ = $(3^4)^{503}$ . Because the units digit of $3^4 \implies 1$ ，so the units digit $1^{503} \implies 1$ . Thus, the units digit of $13^{2012}$ is $\boxed{{\textbf{(A)}\ 1}}$ . ---LarryFlora

@@ Line 4: / Line 4: @@
 <math> \textbf{(A)}\hspace{.05in}1\qquad\textbf{(B)}\hspace{.05in}3\qquad\textbf{(C)}\hspace{.05in}5\qquad\textbf{(D)}\hspace{.05in}7\qquad\textbf{(E)}\hspace{.05in}9 </math>
-==Video Solution==
+== Video Solution by Pi Academy ==
-https://youtu.be/7an5wU9Q5hk?t=1186
+https://youtu.be/7xlBdxcsP3I?si=sFJGKXnBUUN7Su-C
+==Video Solution 2==
+https://youtu.be/6RGNZj0tt2w   ~David
+https://youtu.be/7an5wU9Q5hk?t=1186  ~ pi_is_3.14
+https://youtu.be/6c_s967T7cA ~savannahsolver
 ==Solution 1==
@@ Line 24: / Line 32: @@
 ==Solution 2==
-We find a pattern in the units digit that <math>3^4 \implies 1</math>. We also find <math>2012</math> can be divided by <math>4</math> evenly, meaning <math>2012/4=503</math>. So it gives us the units digit of <math>(3^4)^503</math> is the same as <math>(3^4)</math>. Thus, the answer is <math> \boxed{{\textbf{(A)}\ 1}} </math>.   ---LarryFlora
+Ignore the tens digit of <math>13</math>, we find a pattern in the units digit that <math>3^4 \implies 1</math>. We also find <math>2012</math> can be divided by <math>4</math> evenly, which is <math>2012/4=503</math>. So <math>3^{2012}</math> = <math>(3^4)^{503}</math>. Because the units digit of <math>3^4 \implies 1</math>，so the units digit <math>1^{503} \implies 1</math>. Thus, the units digit of  <math>13^{2012}</math> is <math> \boxed{{\textbf{(A)}\ 1}} </math>.   ---LarryFlora
 ==See Also==
 {{AMC8 box|year=2012|num-b=11|num-a=13}}
 {{MAA Notice}}

2012 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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