`m` is said to be related to `n` if `m` and `n` are integers and `m-n` is divisible by 13. Does this define an equivalence relation?

Explanation

Let us denote this relation by `R`.


Now this relation is reflexive as `(x, x) in R` as `x-x=0` is always divisible by 13 for all `x`.


Similarly the relation is also symmetric as if `(x, y) in R` this gives `(y, x) in R` as if `x-y` is divisible by `13 `then `y-x` is also divisible by 13 for all `x, y`.


Again this relation `R` is also transitive as if `(x, y) in R(y, z) in R` this gives


`(x, z) in R` as `x-y=13 k_{1} ldots . .(1)` [ Since `x-y` is divisible by 13] and `y-z=13 k_{2} ldots .`


(2) [ Since `y` - `z` is also divisible by 13]


Now adding (1) and (2) we get,


`x-z=13(k_{1}+k_{2})`. [For `k_{1}` and `k_{2}` integers ]


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