1983 AIME Problems/Problem 8
What is the largest -digit prime factor of the integer ?
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 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 Since , the largest prime value for is
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