Difference between revisions of "2002 Indonesia MO Problems/Problem 1"
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==Problem== | ==Problem== | ||
− | Show that <math>n^4 - n^2</math> is divisible by <math>12</math> for any integers <math>n > 1</math>. | + | Show that <math>n^4 - n^2</math> is divisible o by <math>12</math> for any integers <math>n > 1</math>. |
==Solution== | ==Solution== | ||
Line 19: | Line 19: | ||
==See Also== | ==See Also== | ||
− | {{Indonesia MO | + | {{Indonesia MO box |
|year=2002 | |year=2002 | ||
|before=First Problem | |before=First Problem | ||
|num-a=2 | |num-a=2 | ||
+ | |eight= | ||
}} | }} | ||
− | [[Category: | + | [[Category:Intermediate Number Theory Problems]] |
Latest revision as of 01:36, 23 November 2021
Problem
Show that is divisible o by for any integers .
Solution
In order for to be divisible by , must be divisible by and .
Lemma 1: is divisible by 4
Note that can be factored into . If is even, then . If , then , and if , then . That means for all positive , is divisible by .
Lemma 2: is divisible by 3
Again, note that can be factored into . If , then . If , then . If , then . That means for all positive , is divisible by .
Because is divisible by and , must be divisible by .
See Also
2002 Indonesia MO (Problems) | ||
Preceded by First Problem |
1 • 2 • 3 • 4 • 5 • 6 • 7 | Followed by Problem 2 |
All Indonesia MO Problems and Solutions |