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Common Multiple from a Cubic (Posted on 2016-05-01) Difficulty: 3 of 5
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?

See The Solution Submitted by Brian Smith    
Rating: 4.0000 (1 votes)

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Start | Comment 1 of 3
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

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