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
[hide]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 |