Is there a relationship between remainders and positive integers?

[Gnoj]

let n be a-positive integer. Prove that a and c leave the same remainder if and only if a - c =nk for some integer k.

 

[Himanshu]

Suppose that

\normalsize A=nQ_{1}+R_{1}

and

\normalsize C=nQ_{2}+R_{2}

substract the two equation to get

\normalsize A-C=n(Q_{1}-Q_{2})+(R_{1}-R_{2})

but the equation

\normalsize A-C=n(Q_{1}-Q_{2})

requires that

\normalsize R_{1}=R_{2}

Therefore the above result.

 

[Gnoj]

Thank You