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 are , and (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 , . However, considering the equation for , testing , we see that always leave a remainder of greater than .
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 |