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A three-digit integer contains one of each of the digits <math> 1</math>, <math> 3</math>, and <math> 5</math>. What is the probability that the integer is divisible by <math> 5</math>?
A three-digit integer contains one of each of the digits <math> 1</math>, <math> 3</math>, and <math> 5</math>. What is the probability that the integer is divisible by <math> 5</math>?


<math> \textbf{(A)}\  \frac{1}{6}  \qquad
<math> \textbf{(A)}\  \frac{1}{6}  \qquad
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\textbf{(D)}\  \frac{2}{3}  \qquad
\textbf{(D)}\  \frac{2}{3}  \qquad
\textbf{(E)}\  \frac{5}{6}</math>
\textbf{(E)}\  \frac{5}{6}</math>
==Solution 1==
The three digit numbers are <math>135,153,351,315,513,531</math>. The numbers that end in <math>5</math> are divisible are <math>5</math>, and the probability of choosing those numbers is <math>\boxed{\textbf{(B)}\ \frac13}</math>.
==Solution 2==
The number is divisible by 5 if and only if the number ends in <math>5</math> (also <math>0</math>, but that case can be ignored, as none of the digits are <math>0</math>)
If we randomly arrange the three digits, the probability of the last digit being <math>5</math> is <math>\boxed{\textbf{(B)}\ \frac13}</math>.
Note: The last sentence is true because there are <math>3</math> randomly-arrangeable numbers)
==See Also==
{{AMC8 box|year=2009|num-b=12|num-a=14}}
{{MAA Notice}}

Latest revision as of 15:32, 14 August 2021

Problem

A three-digit integer contains one of each of the digits $1$, $3$, and $5$. What is the probability that the integer is divisible by $5$?

$\textbf{(A)}\ \frac{1}{6} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{1}{2} \qquad \textbf{(D)}\ \frac{2}{3} \qquad \textbf{(E)}\ \frac{5}{6}$

Solution 1

The three digit numbers are $135,153,351,315,513,531$. The numbers that end in $5$ are divisible are $5$, and the probability of choosing those numbers is $\boxed{\textbf{(B)}\ \frac13}$.

Solution 2

The number is divisible by 5 if and only if the number ends in $5$ (also $0$, but that case can be ignored, as none of the digits are $0$) If we randomly arrange the three digits, the probability of the last digit being $5$ is $\boxed{\textbf{(B)}\ \frac13}$.

Note: The last sentence is true because there are $3$ randomly-arrangeable numbers)

See Also

2009 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
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

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