Difference between revisions of "1983 AIME Problems/Problem 8"
(→Solution 2: Clarification of Solution 1) |
(→Solution 2: Clarification of Solution 1) |
||
Line 5: | Line 5: | ||
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 == |
Revision as of 11:22, 14 August 2019
Problem
What is the largest -digit prime factor of the integer ?
Solution
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 a 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 .
So basically, is the largest prime number such that 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 |