2022/026) Let P(x) be any polynomial with integer coefficients such that P(21)=17,P(32)=−247and P(37)=33. Prove that if P(N)=N+51 for some integer N, then N=26.

Using the property of polynomial that $a-b | p(a)- p(b)$ we have between N and 21

$N-21 | P(N) - p(21)$

As $P(N) = N+ 51$

We get $N-21 | N+ 51 -  17$

or $N-21 | N + 34| or | N - 21 |(N+ 34) - (N-21)$

or $N-21 | 55$

This gives N-21 = 1 or 5 or 11 or 55 or -1 or -5 or - 11 or -55

So N = 26 or 32 or 36 or 76 or 20 or 16 or 10 or - 34 (1)

Now with 32 we get $N- 32 | N + 51 - (-247)$ or $N-32 | 330$ (2)

Finally with 37 we get $N -37 | N + 51 - 33$ | or $N- 37 | 55$

from  set of (1) we get 26 or 32 or 36 

from above (3) values we get   N = 26

so N  = 26 is the answer