P(x) is a monic cubic polynomial with all coefficients positive integers. Also for all integers n, P(n) is a multiple of m.
What is the largest value of m?
At least m=6 as
x^3+3x^2+2x+6 is mod 6 for each x integer value n.
Let be P(x) =x^3+ax^2+bx+c. If for each value n (integer) of x P(n) is mod m, then it should be that
n^3 (mod m) + a*n^2 (mod m) + b*n (mod m) = -c.
Trying some values for m=3, 6, 7, 8, 9 with c=0. I find that for m=6 if a and (b+1) are mod 3, then P(n) is mod 6 for all values of n.
The puzzle states that c is a positive integer (non 0). But no problem here because an addition of a c value do not change solution, if c is also mod 6.
Other way to see it is considering that the product of three consecutive numbers (0 avoided) is always mod 6.
Then n*(n+1)(n+2) is mod 6.
Edited on May 1, 2016, 5:00 pm
Posted by armando
on 2016-05-01 11:57:48