Congruences

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

[itex]a \equiv b \mod m[/itex], means [itex]a[/itex] is a multiple of [itex]m[/itex] plus [itex]b[/itex]. (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

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

I'm sorry, that remark was wrong. Just use the second one:

[itex]a \equiv b \mod m[/itex] 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):

[tex]2 = 3x \mod 5[/tex]

or

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