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Difference between revisions of "2017 AMC 12B Problems/Problem 16"

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Solution by: armang32324
 
Solution by: armang32324
  
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==Solution==
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==Solution 2==
  
 
If <math>21!</math> prime factorizes into <math>p</math> prime factors with exponents <math>e_1</math> through <math>e_k</math>, then the product of the sums of each of these exponents plus <math>1</math> should be over <math>60,000</math>. If <math>e_2</math> is the exponent of <math>2</math> in the prime factorization of 21!, then we can find the number of odd factors of <math>21!</math> by dividing the total by <math>e_2+1</math>. Then, the number of odd divisors over total divisors is <math>\dfrac{1}{e_2+1}</math>. We can find <math>e_2</math> easily using Legendre's, so our final answer is <math>\dfrac{1}{10 + 5 + 2 + 1+1} = \boxed{\text{(B)} \dfrac{1}{19}}.</math>
 
If <math>21!</math> prime factorizes into <math>p</math> prime factors with exponents <math>e_1</math> through <math>e_k</math>, then the product of the sums of each of these exponents plus <math>1</math> should be over <math>60,000</math>. If <math>e_2</math> is the exponent of <math>2</math> in the prime factorization of 21!, then we can find the number of odd factors of <math>21!</math> by dividing the total by <math>e_2+1</math>. Then, the number of odd divisors over total divisors is <math>\dfrac{1}{e_2+1}</math>. We can find <math>e_2</math> easily using Legendre's, so our final answer is <math>\dfrac{1}{10 + 5 + 2 + 1+1} = \boxed{\text{(B)} \dfrac{1}{19}}.</math>

Revision as of 21:02, 30 September 2024

Problem

The number $21!=51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd?

$\textbf{(A)}\ \frac{1}{21} \qquad \textbf{(B)}\ \frac{1}{19} \qquad \textbf{(C)}\ \frac{1}{18} \qquad \textbf{(D)}\ \frac{1}{2} \qquad \textbf{(E)}\ \frac{11}{21}$

Solution 1

We can consider a factor of $21!$ to be odd if it does not contain a $2$; hence, finding the exponent of $2$ in the prime factorization of $21!$ will help us find our answer. We can start off with all multiples of $2$ up to $21$, which is $10$. Then, we find multiples of $4$, which is $5$. Next, we look at multiples of $8$, of which there are $2$. Finally, we know there is only one multiple of $16$ in the set of positive integers up to $21$. Now, we can add all of these to get $10+5+2+1=18$. We know that, in the prime factorization of $21!$, we have $2^{18}$, and the only way to have an odd number is if there is not a $2$ in that number's prime factorization. This only happens with $2^{0}$, which is only one of the 19 different exponents of 2 we could have (of which having each exponent is equally likely). Hence, we have $\boxed{\text{(B)} \dfrac{1}{19}}.$

Solution by: armang32324

Solution 2

If $21!$ prime factorizes into $p$ prime factors with exponents $e_1$ through $e_k$, then the product of the sums of each of these exponents plus $1$ should be over $60,000$. If $e_2$ is the exponent of $2$ in the prime factorization of 21!, then we can find the number of odd factors of $21!$ by dividing the total by $e_2+1$. Then, the number of odd divisors over total divisors is $\dfrac{1}{e_2+1}$. We can find $e_2$ easily using Legendre's, so our final answer is $\dfrac{1}{10 + 5 + 2 + 1+1} = \boxed{\text{(B)} \dfrac{1}{19}}.$

~ icecreamrolls8

Solution

If a factor of $21!$ is odd, that means it contains no factors of $2$. We can find the number of factors of two in $21!$ by counting the number multiples of $2$, $4$, $8$, and $16$ that are less than or equal to $21$ (Legendre's Formula). After some quick counting we find that this number is $10+5+2+1 = 18$. If the prime factorization of $21!$ has $18$ factors of $2$, there are $19$ choices for each divisor for how many factors of $2$ should be included ($0$ to $18$ inclusive). The probability that a randomly chosen factor is odd is the same as if the number of factors of $2$ is $0$ which is $\boxed{\textbf{(B)}\frac{1}{19}}$.

Solution by: vedadehhc

Solution 2

We can write $21!$ as its prime factorization: \[21!=2^{18}\times3^9\times5^4\times7^3\times11\times13\times17\times19\]

Each exponent of these prime numbers are one less than the number of factors at play here. This makes sense; $2^{18}$ is going to have $19$ factors: $2^0, 2^1, 2^2,...\text{ }2^{18}$, and the other exponents will behave identically.

In other words, $21!$ has $(18+1)(9+1)(4+1)(3+1)(1+1)(1+1)(1+1)(1+1)$ factors.

We are looking for the probability that a randomly chosen factor of $21!$ will be odd--numbers that do not contain multiples of $2$ as factors.

From our earlier observation, the only factors of $21!$ that are even are ones with at least one multiplier of $2$, so our probability of finding an odd factor becomes the following: \[P(\text{odd})=\dfrac{\text{number of odd factors}}{\text{number of all factors}}=\dfrac{(9+1)(4+1)(3+1)(1+1)(1+1)(1+1)(1+1)}{(18+1)(9+1)(4+1)(3+1)(1+1)(1+1)(1+1)(1+1)}=\dfrac{1}{(18+1)}=\boxed{\dfrac{1}{19}}\]

Solution submitted by David Kim

Video Solution

https://youtu.be/ZLHNTSIcGM8

-MistyMathMusic

See Also

2017 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 15 		Followed by
Problem 17
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All AMC 12 Problems and Solutions

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