Amazing Detail. Cheers. Just a great feeling seeing someone spend so much time and energy typing this out for others. 
P.S. @anudeep2011:Lucasā Theorem holds true even for prime powers(like 27=3^3) (Called generalized Lucasā theorem)
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Would be great if there was separate thread about this in latex format. Great answer!!
hey can anyone tell me when to use lucas theorem and when to use fermat theorem.
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Lucas n>>p, else fermat.
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