1995 AIME Problems/Problem 6
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Problem
Let How many positive integer divisors of
are less than
but do not divide
?
Solution
We know that must have
factors by its prime factorization. If we group all of these factors (excluding
) into pairs that multiply to
, then one factor per pair is less than
, and so there are
factors of
that are less than
. There are
factors of
, which clearly are less than
, but are still factors of
. Therefore, there are
factors of
that do not divide
.
See also
1995 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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