## Congruences

What exactly does this mean?
a=b (mod m)
My teacher didn't attempt explain this but I'd still like to know.

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 Recognitions: Homework Help Science Advisor $a \equiv b \mod m$, means $a$ is a multiple of $m$ plus $b$. (a,b and m are integers) It means when you divide a by b, you get a rest of m. In algebra, a,b and m are not necessarily integers, but in most cases they are.
 So how do you solve (2/3)= x (mod 5) for x? I'm told that x=4, but if m<|b|, then that can't be right.

Recognitions:

## Congruences

 What exactly does this mean? a=b (mod m)
For integers "a", "b", and "m" (m > 0),
a = b (mod m)
if and only if (a - b) is exactly divisible by "m" (or equivalently, that "a" divided by "m" has the same remainder as "b" divided by "m").

Example:
10 = 4 (mod 3)
---> (10 - 4)/3 = Integer
---> {10 divided by 3} has same remainder as {4 divided by 3}

~~

Recognitions:
Homework Help
 Quote by yay_goobers2112 So how do you solve (2/3)= x (mod 5) for x? I'm told that x=4, but if m<|b|, then that can't be right.
I'm sorry, that remark was wrong. Just use the second one:

$a \equiv b \mod m$ means division of a by b will give rest m.

I'ven't seen fractions on the left side of congruences before, but they probably mean (multiplying both sides by 3):

$$2 = 3x \mod 5$$

or

$$2-3x = 0 \mod 5$$
So 2-3x must be a multiple of 5, so x =4 will work.