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