2017 AMC 10 B
Complete problem set with solutions and individual problem pages
The number , has over positive integer divisors. One of them is chosen at random. What is the probability that it is odd? (2017 AMC 10B Problem, Question#20)
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Solution 1
We note that the only thing that affects the parity of the factor are the powers of . There are factors of in the number. Thus, there are cases in which a factor of would be even (have a factor of in its prime factorization), and case in which a factor of would be odd. Therefore, the answer is .
Solution 2
Consider how to construct any divisor of . First by Legendre's theorem for the divisors of a factorial , we have that there are a total of factors of in the number. can take up either ,,,,, or all factors of , for a total of possible cases. In order for to be odd, however, it must have factors of , meaning that there is a probability of case cases.
