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Question

Prove that the sum of the coefficients of the odd powers of x is the expansion of (1+x+x2+x3+x4)n1, when n is a prime number other than 5, is divisible by n.

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Solution

(1+x+x2+x3+x4)n1=1+c1x+c2x2+c3x3+........(A)(1x+x2x3+x4)n1=1c1x+c2x2c3x3+..........(B)
Putting x=1 and subtracting (i) and (ii)
5n11=2(c1+c3+c5.......)c1+c3+c5......=5n112
Using fermats theorem if N is prime to p then Np11 is divisible by p
5n11 is divisible by n as n is prime other then 5
Hence proved.

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