Difference between revisions of "2006 JBMO Problems/Problem 1"
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Let us define set <math>S = \{1, 2, 3 ... n-1\}</math> | Let us define set <math>S = \{1, 2, 3 ... n-1\}</math> | ||
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<math>Case 1: q > p</math> | <math>Case 1: q > p</math> |
Revision as of 23:11, 26 December 2018
Problem
If is a composite number, then
divides
.
Solution
We shall prove a more stronger result that divides
for any composite number
which will cover the case of problem statement.
Let where
.
Let us define set
First let's note that
Now, all multiples of from
to
Since we have that
Also, since
we have that
So, we have that ,
in other words,
divides
Now, all multiples of from
to
Since we have that
Also, since so we have that
So, we have that ,
in other words,
divides
Thus divides
.