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I would appreciate some help on this simple problem. I'm trying to prove that if n divides a - b, then n divides a and b with the same remainder. All numbers are integers of course. I got...

Hypothesis: [itex]a-b = nm[/itex] for some m. We also have, by the euclid's algorithm, that [itex]\exists ! m_a, r_a[/itex] such that [itex]a=nm_a + r_a[/itex] and [itex]\exists ! m_b, r_b[/itex] such that [itex]b=nm_b+ r_b[/itex]. Combining these 2 results gives the equation [itex]nm_a+r_a-nm_b-r_b=nm[/itex]. From there I don't know what do to.

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# Prove that if n divides a - b, then n divides a and b

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