2017 USAJMO Problems/Problem 4
Problem
Are there any triples of positive integers such that
is prime that properly divides the positive number
?
Let and
. We have that:
Let
. Then
is
, which must be divisible by
.
Since ,
, and since
divides
,
must divide either
or
.
It is easy to see that the minimum of is
. Since
,
cannot divide
, so
must divide
. If
,
. But
, so
. If
is prime (p > 47 + 43 = 90), then
has to be even, making one of
even, making
an even number, which is a contradiction.
Thus, there are no integer triples that work.
~AopsUser101
See also
2017 USAJMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |