I am still confused of how we can expect the order of an modulo multiplication group which has very big orders without arranging each element one by one.
I assume you are referring to finding the order of $ U(750) $. Here is how I did it on the homework last week: Find a formula for $ |U(p^n)| $ where p is a prime. Then, try factoring 750 into two numbers, one of them in a convenient form. You can get $ 750 = 6*5^3 $. Five is prime, and we know $ |U(5^3)| $. We can also get $ |U(6)| $ pretty easily. Then, from the previous homework, we had a conjecture on what, in general, is $ |U(a*b)| $. Use that conjecture with the factors of 750 you generated, and there's your prediction. Of course, if you proved your conjecture, it would cease to be simply a prediction. -Josh Magner
