# Thread: Congruence help

1. ## Congruence help

Let [a] = {x e Z | x and a are congruent mod m}

This is the exercise:

Describe the set [5] if m = 1. Show that [5] = [-1] in this case.

any IDeas?

2. Originally Posted by jzellt
Let [a] = {x e Z | x and a are congruent mod m}
Describe the set [5] if m = 1. Show that [5] = [-1] in this case.
To clarify did you mean,

$[a] = \{x \in \mathbb{Z} | x \equiv a \pmod m\}$

What part are you confused about?

x and a are congruent mod m means the remainder of $\frac{x}{m}$ is the same as the remainder of $\frac{a}{m}$

Because 5 mod 1 = 0, because $\frac{5}{1} = 5$ (with no remainder) we have,
[5] = {all integers that have a zero remainder when divided by 1} = {all integers} = $\mathbb{Z}$

Show that [-1] is the set of all integers too.