Difference between revisions of "2010 AMC 10B Problems/Problem 18"
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==Solution== | ==Solution== | ||
− | First we factor <math>abc+ab+a</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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+ | We see that <math>a</math> is divisible by <math>3</math> with probability <math>\frac{1}{3}</math>. We only need to calculate the probability that <math>bc + b + 1</math> is divisible by <math>3</math>. | ||
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+ | We need <math>bc + b + 1 \equiv 0</math>(mod <math>3</math>) or <math>b(c + 1) \equiv 2</math>(mod <math>3</math>). Using some modular arithmetic, <math>b \equiv 2</math> (mod <math>3</math>) and <math>c \equiv 0</math> (mod <math>3</math>) or <math>b \equiv 1</math> (mod <math>3</math>) and <math>c \equiv 1</math>(mod <math>3</math>). The both cases happen with probability <math>\frac{1}{3} * \frac{1}{3} = \frac{1}{9}</math> so the total probability is <math>\frac{2}{9}</math>. | ||
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+ | Then the answer is <math>\frac{1}{3} + \frac{2}{3} * \frac{2}{9} = \frac{13}{27}</math> or <math>\boxed{E}</math>. | ||
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==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 21:03, 18 January 2015
Problem
Positive integers , , and are randomly and independently selected with replacement from the set . What is the probability that is divisible by ?
Solution
First we factor as , so in order for the number to be divisible by 3, either is divisible by , or is divisible by .
We see that is divisible by with probability . We only need to calculate the probability that is divisible by .
We need (mod ) or (mod ). Using some modular arithmetic, (mod ) and (mod ) or (mod ) and (mod ). The both cases happen with probability so the total probability is .
Then the answer is or .
See Also
2010 AMC 10B (Problems • Answer Key • Resources) | ||
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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