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

Line 2: Line 2:
  
 
<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>
 
<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>
 +
 +
==Solution==
 +
The problem wants us to find the units digit of <math> 13^{2012} </math>, therefore, we can eliminate the tens digit of <math> 13 </math>, because the tens digit will not affect the final result. So our new expression is <math> 3^{2012} </math>. Now we need to look for a pattern in the units digit.
 +
 +
<math> 3^1 \implies 3 </math>
 +
 +
<math> 3^2 \implies 9 </math>
 +
 +
<math> 3^3 \implies 7 </math>
 +
 +
<math> 3^4 \implies 1 </math>
 +
 +
<math> 3^5 \implies 3 </math>
 +
 +
We observe that there is a pattern for the units digit which recurs every four powers of three. Using this pattern, we find that the units digit of <math> 13^{2012} </math> is
 +
<math> \boxed{{\textbf{(A)}\ 1}} </math>.
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2012|num-b=11|num-a=13}}
 
{{AMC8 box|year=2012|num-b=11|num-a=13}}

Revision as of 10:50, 24 November 2012

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

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

Solution

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 find that the units digit of $13^{2012}$ is $\boxed{{\textbf{(A)}\ 1}}$.

See Also

2012 AMC 8 (ProblemsAnswer KeyResources)
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 AJHSME/AMC 8 Problems and Solutions
Invalid username
Login to AoPS