I've encountered what seems to be two different notations for modular arithmetic and I'm confused as to whether they mean the same thing. My abstract algebra textbook (Pinter) and professor would write, for example, 5 = 15mod(10), as though mod(10) is an operation that returns the amount by which 15 differs from a multiple of 10. But the other notation that I've run into, and the one that seems to be more common, is to write 15 ≡ 5mod(10), identifying 15 as an element of the set of integers that are 5 more than a multiple of 10. Which notation is generally preferred? Are there cases where it is more appropriate to use one notation instead of the other?