Difference between revisions of "2003 AMC 10A Problems/Problem 15"
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There are <math>100</math> integers in the set. | There are <math>100</math> integers in the set. | ||
− | Since every | + | Since every <math>2^{\text{nd}}</math> integer is divisible by <math>2</math>, there are <math>\lfloor\frac{100}{2}\rfloor=50</math> integers divisible by <math>2</math> in the set. |
− | To be divisible by both <math>2</math> and <math>3</math>, a number must be divisible by <math> | + | To be divisible by both <math>2</math> and <math>3</math>, a number must be divisible by <math>(2,3)=6</math>. |
− | Since every | + | Since every <math>6^{\text{th}}</math> integer is divisible by <math>6</math>, there are <math>\lfloor\frac{100}{6}\rfloor=16</math> integers divisible by both <math>2</math> and <math>3</math> in the set. |
So there are <math>50-16=34</math> integers in this set that are divisible by <math>2</math> and not divisible by <math>3</math>. | So there are <math>50-16=34</math> integers in this set that are divisible by <math>2</math> and not divisible by <math>3</math>. | ||
− | Therefore, the desired probability is <math>\frac{34}{100}=\frac{17}{50} \Rightarrow C</math> | + | Therefore, the desired probability is <math>\frac{34}{100}=\frac{17}{50}\Rightarrow\boxed{\mathrm{(C)}\ \frac{17}{50}}</math> |
+ | |||
+ | ==Video Solution by WhyMath== | ||
+ | https://youtu.be/UfzS5griBic | ||
+ | |||
+ | ~savannahsolver | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://www.youtube.com/watch?v=4IlfkRW660E ~David | ||
+ | |||
+ | ==Controversy== | ||
+ | Due to the wording of the question, it may be taken as "Find the probability that an integer in said set is divisible by 2 and not 3 EXISTS". One example would be 2, which is not a multiple of 3, thus the probability is 1. But because 1 is not an option, we can assume that it was not meant like that. | ||
== See Also == | == See Also == | ||
− | + | {{AMC10 box|year=2003|ab=A|num-b=14|num-a=16}} | |
− | |||
− | |||
[[Category:Introductory Number Theory Problems]] | [[Category:Introductory Number Theory Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 13:34, 17 August 2023
Problem
What is the probability that an integer in the set is divisible by and not divisible by ?
Solution
There are integers in the set.
Since every integer is divisible by , there are integers divisible by in the set.
To be divisible by both and , a number must be divisible by .
Since every integer is divisible by , there are integers divisible by both and in the set.
So there are integers in this set that are divisible by and not divisible by .
Therefore, the desired probability is
Video Solution by WhyMath
~savannahsolver
Video Solution
https://www.youtube.com/watch?v=4IlfkRW660E ~David
Controversy
Due to the wording of the question, it may be taken as "Find the probability that an integer in said set is divisible by 2 and not 3 EXISTS". One example would be 2, which is not a multiple of 3, thus the probability is 1. But because 1 is not an option, we can assume that it was not meant like that.
See Also
2003 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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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