2010 AIME II Problems/Problem 3
Problem 3
Let be the product of all factors
(not necessarily distinct) where
and
are integers satisfying
. Find the greatest positive integer
such that
divides
.
Solution
In general, there are pairs of integers
that differ by
because we can make
anyway from
to
and make
.
Thus, the product is (some people may recognize it as
.)
When we count the number of factors of , we have 4 groups, factors that are divisible by
at least once, twice, three times and four times.
Number that are divisible by at least once:
Exponent corresponding to each one of them
Sum
Number that are divisible by at least twice:
Exponent corresponding to each one of them
Sum
Number that are divisible by at least three times:
Exponent corresponding to each one of them
Sum
Number that are divisible by at least four times:
Exponent corresponding to each one of them
Sum
summing all this we have
See also
2010 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |