Difference between revisions of "2010 AMC 10B Problems/Problem 18"

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(Solution)
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<math>\textbf{(A)}\ \dfrac{1}{3} \qquad \textbf{(B)}\ \dfrac{29}{81} \qquad \textbf{(C)}\ \dfrac{31}{81} \qquad \textbf{(D)}\ \dfrac{11}{27} \qquad \textbf{(E)}\ \dfrac{13}{27}</math>
 
<math>\textbf{(A)}\ \dfrac{1}{3} \qquad \textbf{(B)}\ \dfrac{29}{81} \qquad \textbf{(C)}\ \dfrac{31}{81} \qquad \textbf{(D)}\ \dfrac{11}{27} \qquad \textbf{(E)}\ \dfrac{13}{27}</math>
  
==Solution==
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==Solution 1==
 
First we factor <math>abc + ab + a</math> as <math>a(bc + b + 1)</math>, so in order for the number to be divisible by 3, either <math>a</math> is divisible by <math>3</math>, or <math>bc + b + 1</math> is divisible by <math>3</math>.  
 
First we factor <math>abc + ab + a</math> as <math>a(bc + b + 1)</math>, so in order for the number to be divisible by 3, either <math>a</math> is divisible by <math>3</math>, or <math>bc + b + 1</math> is divisible by <math>3</math>.  
  
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Then the answer is <math>\frac{1}{3} + \frac{2}{3}\cdot\frac{2}{9} = \frac{13}{27}</math> or <math>\boxed{E}</math>.
 
Then the answer is <math>\frac{1}{3} + \frac{2}{3}\cdot\frac{2}{9} = \frac{13}{27}</math> or <math>\boxed{E}</math>.
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==Solution 2==
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We see that since <math>2010</math> is divisible by <math>3</math>, the probability that any one of <math>a</math>, <math>b</math>, or <math>c</math> being divisible by <math>3</math> is <math>\frac{1}{3}</math>. Because of this, we can shrink the set of possibilities for <math>a</math>, <math>b</math>, and <math>c</math> to the set <math>\{1,2,3\}</math> without affecting the probability in question.
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Listing out all possible combinations for <math>a</math>, <math>b</math>, and <math>c</math>, we see that the answer is <math>\bold{\boxed{\left( E\right) \frac{13}{27}}}</math>.
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2010|ab=B|num-b=17|num-a=19}}
 
{{AMC10 box|year=2010|ab=B|num-b=17|num-a=19}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 19:27, 13 February 2018

Problem

Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3,\dots, 2010\}$. What is the probability that $abc + ab + a$ is divisible by $3$?

$\textbf{(A)}\ \dfrac{1}{3} \qquad \textbf{(B)}\ \dfrac{29}{81} \qquad \textbf{(C)}\ \dfrac{31}{81} \qquad \textbf{(D)}\ \dfrac{11}{27} \qquad \textbf{(E)}\ \dfrac{13}{27}$

Solution 1

First we factor $abc + ab + a$ as $a(bc + b + 1)$, so in order for the number to be divisible by 3, either $a$ is divisible by $3$, or $bc + b + 1$ is divisible by $3$.

We see that $a$ is divisible by $3$ with probability $\frac{1}{3}$. We only need to calculate the probability that $bc + b + 1$ is divisible by $3$.

We need $bc + b + 1 \equiv 0\pmod 3$ or $b(c + 1) \equiv 2\pmod 3$. Using some modular arithmetic, $b \equiv 2\pmod 3$ and $c \equiv 0\pmod 3$ or $b \equiv 1\pmod 3$ and $c \equiv 1\pmod 3$. The both cases happen with probability $\frac{1}{3} * \frac{1}{3} = \frac{1}{9}$ so the total probability is $\frac{2}{9}$.

Then the answer is $\frac{1}{3} + \frac{2}{3}\cdot\frac{2}{9} = \frac{13}{27}$ or $\boxed{E}$.


Solution 2

We see that since $2010$ is divisible by $3$, the probability that any one of $a$, $b$, or $c$ being divisible by $3$ is $\frac{1}{3}$. Because of this, we can shrink the set of possibilities for $a$, $b$, and $c$ to the set $\{1,2,3\}$ without affecting the probability in question.

Listing out all possible combinations for $a$, $b$, and $c$, we see that the answer is $\bold{\boxed{\left( E\right) \frac{13}{27}}}$.

See Also

2010 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 17
Followed by
Problem 19
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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