Difference between revisions of "1983 AIME Problems/Problem 8"
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== Problem == | == Problem == | ||
What is the largest <math>2</math>-digit [[prime]] factor of the integer <math>n = {200\choose 100}</math>? | What is the largest <math>2</math>-digit [[prime]] factor of the integer <math>n = {200\choose 100}</math>? | ||
| − | == Solution 1 == | + | ==Solution== |
| + | === Solution 1 === | ||
Expanding the [[combination|binomial coefficient]], we get <math>{200 \choose 100}=\frac{200!}{100!100!}</math>. Let the required prime be <math>p</math>; then <math>10 \le p < 100</math>. If <math>p > 50</math>, then the factor of <math>p</math> appears twice in the denominator. Thus, we need <math>p</math> to appear as a factor at least three times in the numerator, so <math>3p<200</math>. The largest such prime is <math>\boxed{061}</math>, which is our answer. | Expanding the [[combination|binomial coefficient]], we get <math>{200 \choose 100}=\frac{200!}{100!100!}</math>. Let the required prime be <math>p</math>; then <math>10 \le p < 100</math>. If <math>p > 50</math>, then the factor of <math>p</math> appears twice in the denominator. Thus, we need <math>p</math> to appear as a factor at least three times in the numerator, so <math>3p<200</math>. The largest such prime is <math>\boxed{061}</math>, which is our answer. | ||
| − | == Solution 2: Clarification of Solution 1 == | + | === Solution 2: Clarification of Solution 1 === |
We know that | We know that | ||
Revision as of 19:23, 14 January 2023
Problem
What is the largest 
-digit prime factor of the integer 
?
Solution
Solution 1
Expanding the binomial coefficient, we get 
. Let the required prime be 
; then 
. If 
, then the factor of appears twice in the denominator. Thus, we need to appear as a factor at least three times in the numerator, so 
. The largest such prime is 
, which is our answer.
Solution 2: Clarification of Solution 1
We know that
![\[{200\choose100}=\frac{200!}{100!100!}\]](http://latex.artofproblemsolving.com/c/a/c/cac11b8537859d11a4e3be8efcef3a4a67c333af.png)
Since 
, there is at least 
factor of in each of the 
in the denominator. Thus there must be at least 
factors of in the numerator 
for to be a factor of 
. (Note that here we assume the minimum because as goes larger in value, the number of factors of in a number decreases,)
So basically, is the largest prime number such that
![\[\left \lfloor\frac{200}{p}\right \rfloor>3\]](http://latex.artofproblemsolving.com/d/6/b/d6b4bcf5fed67b290052cae032950a11ab1ad2be.png)
Since 
, the largest prime value for is 
~ 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 | ||
