2013 Indonesia MO Problems/Problem 4
Problem
Suppose is a prime number and
Prove that
is divisible by
.
Solution
if you let be constant, you can think of it as summing for each
,
, and since its for all
you can add another sum to get
, and for all
we can add another sum, to get
since it has a factor pf
, we need to prove
is always an integer, for prime
, so let
and
for the first case, you get
and for the second you get
, notice how both of these are always integers, thus it is proven
divides
See Also
2013 Indonesia MO (Problems) | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Problem 5 |
All Indonesia MO Problems and Solutions |