Difference between revisions of "1971 Canadian MO Problems/Problem 6"
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Thus, for any integer <math>n</math>, <math>n^2+2n+12</math> is not a multiple of <math>121</math>. | Thus, for any integer <math>n</math>, <math>n^2+2n+12</math> is not a multiple of <math>121</math>. | ||
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== See Also == | == See Also == | ||
{{Old CanadaMO box|num-b=5|num-a=7|year=1971}} | {{Old CanadaMO box|num-b=5|num-a=7|year=1971}} | ||
[[Category:Intermediate Algebra Problems]] | [[Category:Intermediate Algebra Problems]] |
Revision as of 18:18, 7 August 2016
Contents
Problem
Show that, for all integers , is not a multiple of .
Solutions
Solution 1
. 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 .
See Also
1971 Canadian MO (Problems) | ||
Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • | Followed by Problem 7 |