2005 AMC 10A Problems/Problem 21
Problem
For how many positive integers does
evenly divide
?
Solution
If evenly divides
, then
is an integer.
Since we may substitute the RHS in the above fraction.
So the problem asks us for how many positive integers is
an integer.
is an integer when
is a factor of
.
The factors of are
,
,
,
,
, and
.
So the possible values of are
,
,
,
,
, and
.
But isn't a positive integer, so only
,
,
,
, and
are possible values of
.
Therefore the number of possible values of is