1971 Canadian MO Problems/Problem 6
Problem
Show that, for all integers ,
is not a multiple of
.
Solution
. Consider this equation mod 11.
.
The quadratic residues mod 11 are 1, 3, 4, 5, 9, and 0 (as shown below).
If ,
, thus not a multiple of 11, nor 121.
If ,
, thus not a multiple of 11, nor 121.
If ,
, thus not a multiple of 11, nor 121.
If ,
, thus not a multiple of 11, nor 121.
If ,
, thus not a multiple of 11, nor 121.
If ,
, thus not a multiple of 11, nor 121.
If ,
, thus not a multiple of 11, nor 121.
If ,
, thus not a multiple of 11, nor 121.
If ,
, thus not a multiple of 11, nor 121.
If ,
, thus not a multiple of 11, nor 121.
If ,
, thus a multiple of 11. However, considering the equation
,
, thus not a multiple of 121, even though it is a multiple of 11.
Thus, for any integer ,
is not a multiple of
.
1971 Canadian MO (Problems) | ||
Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • | Followed by Problem 7 |