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

Revision as of 12:22, 21 August 2021

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

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

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 $3$ power of 2012 = $(3^4)$ power of $503$ . Because the units digit of $3^4 \implies 1$ ，and the units digit $1$ of power $503 \implies 1$ . Thus, the units digit of $13^{2012}$ is $\boxed{{\textbf{(A)}\ 1}}$ . ---LarryFlora

2012 AMC 8 (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 AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 24: / Line 24: @@
 ==Solution 2==
-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, meaning <math>2012/4=503</math>. So <math>3</math> power of 2012 = <math>(3^4)</math> power of <math>503</math>. Because the units digit of <math>3^4 \implies 1</math>，and the units digit <math>1</math> of power <math>503 \implies 1</math>. Thus, the units digit of  <math>13^{2012}</math> 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</math> power of 2012 = <math>(3^4)</math> power of <math>503</math>. Because the units digit of <math>3^4 \implies 1</math>，and the units digit <math>1</math> of power <math>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}}

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