2007 iTest Problems/Problem 44
Problem
A positive integer between and inclusive is selected at random. If and are natural numbers such that is the probability that and are relatively prime, find the value of .
Solution
Factoring results in , and factoring results in . In order for and to be relatively prime, then can not have multiples of or , and can not be away from a multiple of , so use complementary counting.
There are numbers in the range that are a multiple of , and there are numbers from to that are multiples of or away from a multiple of . However, there are numbers that are a multiple of and a multiple of or six away from a multiple of . Using PIE, there are a total of values of that do not work.
That means the number of values of that work is , and since a number is picked at random from values, the probability that and are relatively prime is . Thus, .
See Also
2007 iTest (Problems, Answer Key) | ||
Preceded by: Problem 43 |
Followed by: Problem 45 | |
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