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For
consider the rational numberWhere
and are positive integers. Prove that if p is an odd prime , then the numberator of is divisible by p.Numbers are the essence of the Universe
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I thought
then I don't know what to do
Last edited by Stanley_Marsh (2007-03-11 14:02:39)
Numbers are the essence of the Universe
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You cant do it that way h[sub]p−1[/sub] is not an integer.
Last edited by JaneFairfax (2007-03-11 15:29:42)
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Its actually really simple. Instead of taking an odd prime p, well take any odd number n = 2a+1 > 1.
Now we rewrite the terms in the second half-sum backwards. Note that 2a = n−1, 2a−1 = n−2 a+1 = n−a.
Note that for all k = 1, 2 a
Therefore
for k = 1, 2 a. This shows that u[sub]n−1[/sub] is divisible by n (so the result is true for all odd integers greater than 1, not just odd primes).
Last edited by JaneFairfax (2007-03-11 17:22:15)
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