2013 Indonesia MO Problems/Problem 6
Problem
A positive integer is called "strong" if there exists a positive integer
such that
is divisible by
.
a. Prove that is strong.
b. If is strong, determine the smallest
(in terms of
) such that
is divisible by
.
Solution
a. Take , notice how
is odd,
so its divisible
b. Notice how is always odd as if it is even then even+odd=odd and cant be divisible by 2^m, and
is always odd as if it is even
,
which is not true because if m=1 then it is odd. By LTE,
as it is divisible by 2^m, and the smallest
is
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 |