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== | ||
| − | In order for <math>n^4 - n^2</math> to be divisible by <math>12</math>, <math>n^2 | + | In order for <math>n^4 - n^2</math> to be divisible by <math>12</math>, <math>n^4 - n^2</math> must be divisible by <math>4</math> and <math>3</math>. |
<br> | <br> | ||
| − | '''Lemma 1: | + | '''Lemma 1: <math>n^4 - n^2</math> is divisible by 4'''<br> |
Note that <math>n^4 - n^2</math> can be factored into <math>n^2 (n+1)(n-1)</math>. If <math>n</math> is even, then <math>n^2 \equiv 0 \pmod{4}</math>. If <math>n \equiv 1 \pmod{4}</math>, then <math>n-1 \equiv 0 \pmod{4}</math>, and if <math>n \equiv 3 \pmod{4}</math>, then <math>n+1 \equiv 0 \pmod{4}</math>. That means for all positive <math>n</math>, <math>n^2 (n+1)(n-1)</math> is divisible by <math>4</math>. | Note that <math>n^4 - n^2</math> can be factored into <math>n^2 (n+1)(n-1)</math>. If <math>n</math> is even, then <math>n^2 \equiv 0 \pmod{4}</math>. If <math>n \equiv 1 \pmod{4}</math>, then <math>n-1 \equiv 0 \pmod{4}</math>, and if <math>n \equiv 3 \pmod{4}</math>, then <math>n+1 \equiv 0 \pmod{4}</math>. That means for all positive <math>n</math>, <math>n^2 (n+1)(n-1)</math> is divisible by <math>4</math>. | ||
<br> | <br> | ||
| − | '''Lemma 2: | + | '''Lemma 2: <math>n^4 - n^2</math> is divisible by 3'''<br> |
Again, note that <math>n^4 - n^2</math> can be factored into <math>n^2 (n+1)(n-1)</math>. If <math>n \equiv 0 \pmod{3}</math>, then <math>n^2 \equiv 0 \pmod{3}</math>. If <math>n \equiv 1 \pmod{3}</math>, then <math>n-1 \equiv 0 \pmod{3}</math>. If <math>n \equiv 2 \pmod{3}</math>, then <math>n+1 \equiv 0 \pmod{3}</math>. That means for all positive <math>n</math>, <math>n^2 (n+1)(n-1)</math> is divisible by <math>3</math>. | Again, note that <math>n^4 - n^2</math> can be factored into <math>n^2 (n+1)(n-1)</math>. If <math>n \equiv 0 \pmod{3}</math>, then <math>n^2 \equiv 0 \pmod{3}</math>. If <math>n \equiv 1 \pmod{3}</math>, then <math>n-1 \equiv 0 \pmod{3}</math>. If <math>n \equiv 2 \pmod{3}</math>, then <math>n+1 \equiv 0 \pmod{3}</math>. That means for all positive <math>n</math>, <math>n^2 (n+1)(n-1)</math> is divisible by <math>3</math>. | ||
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==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 00:36, 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 | ||
