Difference between revisions of "1983 AIME Problems/Problem 8"
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== Problem == | == Problem == | ||
| + | What is the largest 2-digit prime factor of the integer <math>\binom{200}{100}</math>? | ||
== Solution == | == Solution == | ||
| + | Expanding the [[binomial coefficient]], we get <math>{200 \choose 100}=\frac{200!}{100!100!}</math>. | ||
| + | |||
| + | Therefore, our two digit [[prime]] <math>p</math> must satisfy <math>3p<200</math>. The largest such prime is <math>61</math>, which is our answer. | ||
| + | |||
| + | ---- | ||
| + | |||
| + | * [[1983 AIME Problems/Problem 7|Previous Problem]] | ||
| + | * [[1983 AIME Problems/Problem 9|Next Problem]] | ||
| + | * [[1983 AIME Problems|Back to Exam]] | ||
== See also == | == See also == | ||
| − | * [[ | + | * [[AIME Problems and Solutions]] |
| + | * [[American Invitational Mathematics Examination]] | ||
| + | * [[Mathematics competition resources]] | ||
| + | |||
| + | [[Category:Intermediate Combinatorics Problems]] | ||
Revision as of 23:06, 23 July 2006
Problem
What is the largest 2-digit prime factor of the integer 
?
Solution
Expanding the binomial coefficient, we get 
.
Therefore, our two digit prime 
must satisfy 
. The largest such prime is 
, which is our answer.
