IIT JEE , AIEEE Forum - Mathematics , Algebra

hsbhatt

We again use the result that P(a) - P(b) is div by (a-b)

Now, if an integer N is div by c then N = kc

We must have (kc-c)| P(kc) - P(c) So if P(N) = 0, then P(c) is div by c which is a contradiction.

Now, suppose N is not a multiple of c. Then N = cd+r where o<r<c

Then N-r|P(N) - P(r) or cd|P(N)-P(r). So, if P(N) = 0, P(r) is divisible by c which is a contradiction as P(k) is not divisible by c for k<=c.
Hence P(x) = 0 has no solution in integers

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