2020 AMC 10A Problems/Problem 22
Problem
For how many positive integers is
not divisible by
? (Recall that
is the greatest integer less than or equal to
.)
Solution 1
Let . If the expression
is not divisible by
, then the three terms in the expression must be
, which would imply that
is a divisor of
but not
, or
, which would imply that
is a divisor of
but not
.
has
factors, and
has
factors. However,
does not work because
a divisor of both
and
, and since
is counted twice, the answer is
.
Solution 2 (Casework)
Let . Notice that for every integer
, if
is an integer, then the three terms in the expression must be
, if
is an integer, then the three terms in the expression must be
, and if
is an integer, then the three terms in the expression must be
. This is due to the fact that 998, 999, and 1000 share no factors other than 1.
So, there are two cases:
Case 1: divides
Because divides
, the number of possibilities for
is the same as the number of factors of
, excluding
.
=
So, the total number of factors of
is
.
However, we have to subtract
, because we have counted
when
\neq
.
Video Solution
~IceMatrix
See Also
2020 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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All AMC 10 Problems and Solutions |
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