2017 AMC 12B Problems/Problem 16
Problem 16
The number has over
positive integer divisors. One of them is chosen at random. What is the probability that it is odd?
Solution
If a factor of is odd, that means it contains no factors of
. We can find the number of factors of two in
by counting the number multiples of
,
,
, and
that are less than or equal to
.After some quick counting we find that this number is
. If the prime factorization of
has
factors of
, there are
choices for each divisor for how many factors of
should be included (
to
inclusive). The probability that a randomly chosen factor is odd is the same as if the number of factors of
is
which is
.
Solution by: vedadehhc