When is m33 divisible by 36? (m is an integer).

[Silversonic]

1. The problem statement, all variables and given/known data

Work out the order of the following elements;

33 \in Z_{36}


3. The attempt at a solution

It's probably really simple. But this only happens when an integer times 33 is divisible by 36.

That is;

33n = 36m

Which I can re-arrange to find

n = 36m/33

Now, I can keep adding 36/33 in my calculator until I get an integer result, but surely there is an easier way? For example;

33 \equiv -3mod36

Which suggests that 36/3 = 12 is the order of 33. This is fine and dandy, but what happens when I get to a question like, find the order of

15 \in Z_{36}

Then

15 \equiv -21mod36

Which means I'm back to the same problem again. My modular arithmetic is fairly poor, so how would I work this out?

[LCKurtz]

Factor 36 into prime factors and think about what factors k must have for 36 to divide 33k.

[Silversonic]

36 = 3x3x2x2

33 = 11x3

So k must be 3x2x2 = 12.

Also

15 = 5x3

So k must be 3x2x2 = 12.


Thanks!