Thread: Congruence/Residue Class

Q:
Let be a polynomial over Z
Show that if r consecutive vales of f (i.e. values for consecutive integers) are all divisible by r, then r|f(m) for all Z

the remainder of the r consecutive integer must be 0,1,...,r-1.
since , , since

Cool, thx for the response

A few questions:
1. I believe I came up with something similar to the proof you gave but I was wondering if it is required to prove that m mod r {0, 1, ... ,r-1} or is it obvious?

What I am thinking is that that since we are working with mod r and all numbers should be represented by another number mod r then it is safe to assume m mod r {0, 1, ... ,r-1}. Is that correct?

2. In order for this to be true, what does r have to equal? A second part to this question says that this is true with r > 1 and . The problem I see with this is the component of the function

Originally Posted by dlbsd

Cool, thx for the response

A few questions:
1. I believe I came up with something similar to the proof you gave but I was wondering if it is required to prove that m mod r {0, 1, ... ,r-1} or is it obvious?

What I am thinking is that that since we are working with mod r and all numbers should be represented by another number mod r then it is safe to assume m mod r {0, 1, ... ,r-1}. Is that correct?

2. In order for this to be true, what does r have to equal? A second part to this question says that this is true with r > 1 and . The problem I see with this is the component of the function

let , then ,then obviously,