What Does a ≡ b (mod m) Imply in Mathematics?

[yay_goobers2112]

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

 

[Galileo]

$a \equiv b \mod m$, 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.

 

[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.

 

[xanthym]

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}


~~

 

[Galileo]

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.