Modular multiplicative inverse

I don't quite understand this concept, or how to write it down algebraically (not sure what the triple equals thing means).

So let's say I want to find the multiplicative inverse of $\displaystyle 3 \mod{11}$. Am I right in thinking that I am looking for a number that when multiplied by 3, and then divided by 11, I will get remainder 1. The reason I struggle writing this down is because the quotient is irrelevant.

Anyway, to work this out in my head, I would use trial and error:

$\displaystyle 1 \times 3 \mod{11} = 3$

$\displaystyle 2 \times 3 \mod{11} = 6$

$\displaystyle 3 \times 3 \mod{11} = 9$

$\displaystyle 4 \times 3 \mod{11} = 1$ because 12/11 is 1 remainder 1

So the answer is 4.

Presuming what I have done so far is right, I don't understand why Wikipedia says 15 is also an answer? 15*4 = 60, and 60/11 is 5 r5, NOT r1.