Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus
During AMC testing, the AoPS Wiki is in read-only mode and no edits can be made.

Difference between revisions of "2006 JBMO Problems/Problem 1"

(→‎Problem)
(Add →‎See Also)
 
Line 1: Line 1:
−
==Problem==
+
== Problem ==
  
 
If <math>n>4</math> is a composite number, then <math>2n</math> divides <math>(n-1)!</math>.
 
If <math>n>4</math> is a composite number, then <math>2n</math> divides <math>(n-1)!</math>.
Line 39: Line 39:
 
Thus <math>4n</math> divides <math>(n-1)!</math>.
 
Thus <math>4n</math> divides <math>(n-1)!</math>.
  
 +
~Kris17
  
−
<math>Kris17</math>
+
== See Also ==
  
−
=Solution 2=
+
* [[JBMO]]

Latest revision as of 15:57, 28 February 2025

Problem

If $n>4$ is a composite number, then $2n$ divides $(n-1)!$.

Solution

We shall prove a more stronger result that $4n$ divides $(n-1)!$ for any composite number $n>4$ which will cover the case of problem statement.

Let $n = p.q$ where $q \ge p \ge 2$.

Let us define set $S = \{1, 2, 3 ... n-1\}$


$Case 1: q > p$

First let's note that $p, q \in S$

Now, all multiples of $p$ from $p.1$ to $p.(q-1) \in S$

Since $q > p => q > 2 => q-1 \ge 2$ we have that $p.2 \in S$ Also, since $p \ge 2$ we have that $2 \in S$

So, we have that $2, p.2, q \in S$, in other words, $4.p.q$ divides $(n-1)!$


$Case 2: q = p$

Now, all multiples of $p$ from $p.1$ to $p.(p-1) \in S$

Since $p > 2 => p-1 \ge 2$ we have that $p.2 \in S$

Also, since $n = p^2 > 4 => p > 2$ so we have that $2 \in S$

So, we have that $2, p.1, p.2 \in S$, in other words, $4.p^2$ divides $(n-1)!$


Thus $4n$ divides $(n-1)!$.

~Kris17

See Also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2006_JBMO_Problems/Problem_1&oldid=244025"