Difference between revisions of "2023 AMC 10A Problems/Problem 14"
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+ | ==Problem== | ||
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A number is chosen at random from among the first <math>100</math> positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by <math>11</math>? | A number is chosen at random from among the first <math>100</math> positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by <math>11</math>? | ||
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==Solution 1== | ==Solution 1== | ||
− | Among the first <math>100</math> positive integers, there are 9 multiples of 11. We can now perform a little casework on the probability of choosing a divisor which is a multiple of 11 for each of these 9, and see that the probability is 1/2 for each. The probability of choosing these 9 multiples in the first place is <math>\frac{9}{100}</math>, so the final probability is <math>\frac{9}{100} \cdot \frac{1}{2} = \frac{9}{200}</math>, so the answer is <math>\boxed{B}.</math> | + | In order for the divisor chosen to be a multiple of <math>11</math>, the original number chosen must also be a multiple of <math>11</math>. Among the first <math>100</math> positive integers, there are 9 multiples of 11; 11, 22, 33, 44, 55, 66, 77, 88, 99. We can now perform a little casework on the probability of choosing a divisor which is a multiple of 11 for each of these 9, and see that the probability is 1/2 for each. The probability of choosing these 9 multiples in the first place is <math>\frac{9}{100}</math>, so the final probability is <math>\frac{9}{100} \cdot \frac{1}{2} = \frac{9}{200}</math>, so the answer is <math>\boxed{\textbf{(B)}~\frac{9}{200}}.</math> |
+ | |||
+ | <math>11 = 1, 11 \Rightarrow \frac{1}{2}\\ | ||
+ | 22 = 2 \times 11: 1, 2, 11, 22 \Rightarrow \frac{1}{2}\\ | ||
+ | 33 = 3 \times 11: 1, 3, 11, 33 \Rightarrow \frac{1}{2}\\ | ||
+ | 44 = 2^2 \times 11: 1, 2, 4, 11, 22, 44 \Rightarrow \frac{1}{2}\\ | ||
+ | 55 = 5 \times 11: 1, 5, 11, 55 \Rightarrow \frac{1}{2}\\ | ||
+ | 66 = 2 \times 3 \times 11: 1, 2, 3, 6, 11, 22, 33, 66 \Rightarrow \frac{1}{2}\\ | ||
+ | 77 = 7 \times 11: 1, 7, 11, 77 \Rightarrow \frac{1}{2}\\ | ||
+ | 88 = 2^3 \times 11: 1, 2, 4, 8, 11, 22, 44, 88 \Rightarrow \frac{1}{2}\\ | ||
+ | 99 = 3^2 \times 11: 1, 3, 9, 11, 33, 99 \Rightarrow \frac{1}{2}</math> | ||
+ | |||
+ | ~vaisri | ||
+ | |||
+ | ~walmartbrian | ||
+ | |||
+ | ~Shontai | ||
+ | |||
+ | ~(minor formatting changes by Marshall_Huang) | ||
+ | |||
+ | ~Formatting changes to avoid confusion by SHREYU.MEDATATI | ||
+ | |||
+ | ==Solution 2== | ||
+ | |||
+ | There are <math>\left\lfloor\frac{100}{11}\right\rfloor = 9</math> multiples of <math>11</math> under <math>100</math>. Because all of these numbers are multiples of <math>11</math> to the first power and first power only, their factors can either have <math>11</math> as a factor (<math>11^{1}</math>) or not have <math>11</math> as a factor (<math>11^{0}</math>), resulting in a <math>\frac{1}{2}</math> chance of a factor chosen being divisible by <math>11</math>. The chance of choosing any factor of <math>11</math> under <math>100</math> is <math>\frac{9}{100}</math>, so the final answer is <math>\frac{9}{100} \cdot \frac{1}{2} = \boxed{\textbf{(B)}~\frac{9}{200}}.</math> | ||
+ | |||
+ | ~Failure.net | ||
+ | |||
+ | ==Solution 3 (generalized)== | ||
+ | Let <math>N</math> be the positive integer in question. Since <math>N</math> is a multiple of <math>11</math>, we can write <math>N = 11^{e_1}p_1^{e_2}\cdots p_{k-1}^{e_k},</math> the prime factorization of <math>N</math>. To find the total number of divisors divisible by <math>11</math>, observe that there are <math>(e_2 + 1)(e_3 + 1)\cdots (e_k + 1)</math> divisors not divisible by <math>11</math>. For each power of <math>11</math> greater than <math>1</math>, of which there are <math>e_1</math>, it can be paired with any of these other divisors. Therefore, there are | ||
+ | <cmath>e_1(e_2+1)(e_3 + 1)\cdots (e_k + 1)</cmath> | ||
+ | divisors of <math>N</math> that are divisible by <math>11</math>. The total number of divisors of <math>N</math> is <cmath>(e_1 + 1)(e_2 + 1)\cdots (e_k + 1),</cmath> so the probability of choosing a divisor that is divisible by <math>11</math> is | ||
+ | <cmath>\frac{e_1(e_2+1)(e_3 + 1)\cdots (e_k + 1)}{(e_1 + 1)(e_2 + 1)\cdots (e_k + 1)} = \frac{e_1}{e_1 + 1}.</cmath> | ||
+ | For each positive integer less than or equal to <math>100</math>, the highest power of <math>11</math> that divides it is <math>11</math>, so <math>e_1 = 1</math>, meaning that the probability of choosing a divisor (that is divisible by <math>11</math>) of a fixed <math>N</math> is <math>\frac{1}{2}.</math> The probability of choosing any <math>N</math> from the first <math>100</math> positive integers is <math>\frac{1}{100},</math> so the probability of choosing any of these divisors is <math>\frac{1}{100}\cdot \frac{1}{2}.</math> There are <math>9</math> multiples of <math>11</math> less than or equal to <math>100</math>, so the total probability is | ||
+ | <cmath>9\cdot \frac{1}{100}\cdot \frac{1}{2} = \boxed{\textbf{(B)}\ \frac{9}{200}}.</cmath> | ||
+ | |||
+ | -Benedict T (countmath1) | ||
+ | |||
+ | |||
+ | ==Video Solution ⚡️ 1 min solution ⚡️ == | ||
+ | |||
+ | https://youtu.be/CP0Arc_82Y8 | ||
+ | |||
+ | <i> ~Education, the Study of Everything </i> | ||
+ | |||
+ | ==Video Solutions== | ||
+ | https://www.youtube.com/watch?v=jkfsBYzBJbQ | ||
+ | |||
+ | |||
+ | https://www.youtube.com/watch?v=uaf46N6qP54 | ||
+ | |||
+ | ==Video Solution== | ||
+ | |||
+ | https://youtu.be/O_P3E9hDrYY | ||
+ | |||
+ | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ||
+ | |||
+ | |||
+ | https://youtu.be/N2lyYRMuZuk?si=-CyrdswJABoMrnWk&t=825 | ||
+ | ~Math-X | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC10 box|year=2023|ab=A|num-b=13|num-a=15}} | ||
+ | {{MAA Notice}} |
Latest revision as of 23:27, 21 August 2024
Contents
Problem
A number is chosen at random from among the first positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by
?
Solution 1
In order for the divisor chosen to be a multiple of , the original number chosen must also be a multiple of
. Among the first
positive integers, there are 9 multiples of 11; 11, 22, 33, 44, 55, 66, 77, 88, 99. We can now perform a little casework on the probability of choosing a divisor which is a multiple of 11 for each of these 9, and see that the probability is 1/2 for each. The probability of choosing these 9 multiples in the first place is
, so the final probability is
, so the answer is
~vaisri
~walmartbrian
~Shontai
~(minor formatting changes by Marshall_Huang)
~Formatting changes to avoid confusion by SHREYU.MEDATATI
Solution 2
There are multiples of
under
. Because all of these numbers are multiples of
to the first power and first power only, their factors can either have
as a factor (
) or not have
as a factor (
), resulting in a
chance of a factor chosen being divisible by
. The chance of choosing any factor of
under
is
, so the final answer is
~Failure.net
Solution 3 (generalized)
Let be the positive integer in question. Since
is a multiple of
, we can write
the prime factorization of
. To find the total number of divisors divisible by
, observe that there are
divisors not divisible by
. For each power of
greater than
, of which there are
, it can be paired with any of these other divisors. Therefore, there are
divisors of
that are divisible by
. The total number of divisors of
is
so the probability of choosing a divisor that is divisible by
is
For each positive integer less than or equal to
, the highest power of
that divides it is
, so
, meaning that the probability of choosing a divisor (that is divisible by
) of a fixed
is
The probability of choosing any
from the first
positive integers is
so the probability of choosing any of these divisors is
There are
multiples of
less than or equal to
, so the total probability is
-Benedict T (countmath1)
Video Solution ⚡️ 1 min solution ⚡️
~Education, the Study of Everything
Video Solutions
https://www.youtube.com/watch?v=jkfsBYzBJbQ
https://www.youtube.com/watch?v=uaf46N6qP54
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
https://youtu.be/N2lyYRMuZuk?si=-CyrdswJABoMrnWk&t=825
~Math-X
See Also
2023 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.