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Difference between revisions of "1983 AIME Problems/Problem 8"

(→‎Solution 2: Clarification of Solution 1)
(→‎Solution 2: Clarification of Solution 1)
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We know that  
 
We know that  
 
<cmath>{200\choose100}=\frac{200!}{100!100!}</cmath>
 
<cmath>{200\choose100}=\frac{200!}{100!100!}</cmath>
−
Since <math>p<100</math>, there is at least <math>1</math> factor of <math>p</math> in each of the <math>100!</math> in the denominator. Thus there must be at least <math>3</math> factors of <math>p</math> in the numerator <math>200!</math> for <math>p</math> to be a factor of <math>n=\frac{200!}{100!100!}</math>.
+
Since <math>p<100</math>, there is at least <math>1</math> factor of <math>p</math> in each of the <math>100!</math> in the denominator. Thus there must be at least <math>3</math> factors of <math>p</math> in the numerator <math>200!</math> for <math>p</math> to be a factor of <math>n=\frac{200!}{100!100!}</math>. (Note that here we assume the minimum because as <math>p</math> goes larger in value, the number of factors of <math>p</math> in a number decreases,)
  
 
So basically, <math>p</math> is the largest prime number such that
 
So basically, <math>p</math> is the largest prime number such that

Revision as of 11:24, 14 August 2019

Problem

What is the largest $2$-digit prime factor of the integer $n = {200\choose 100}$?

Solution

Expanding the binomial coefficient, we get ${200 \choose 100}=\frac{200!}{100!100!}$. Let the required prime be $p$; then $10 \le p < 100$. If $p > 50$, then the factor of $p$ appears twice in the denominator. Thus, we need $p$ to appear as a factor at least three times in the numerator, so $3p<200$. The largest such prime is $\boxed{061}$, which is our answer.


Solution 2: Clarification of Solution 1

We know that \[{200\choose100}=\frac{200!}{100!100!}\] Since $p<100$, there is at least $1$ factor of $p$ in each of the $100!$ in the denominator. Thus there must be at least $3$ factors of $p$ in the numerator $200!$ for $p$ to be a factor of $n=\frac{200!}{100!100!}$. (Note that here we assume the minimum because as $p$ goes larger in value, the number of factors of $p$ in a number decreases,)

So basically, $p$ is the largest prime number such that \[\lfloor\frac{200}{3}\rfloor>3\] Since $p<\frac{200}{3}\approx66.66...$, the largest prime value for $p$ is $p=61$

~ Nafer

See Also

1983 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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