Q1) Are the GCD, Greatest Common Factor (GCF) and Highest Common Factor (HCF) all referring to the same thing?

I'm using the GCD to simplify fractions to their lowest terms. To find the GCD I've been using two methods. The first method is doing prime factorization and multiplying the common prime factors together. The second method is called Euclid's algorithm. Both methods seem slow.

Q2) Should I be using GCD to simplify fractions? And does using prime factorization have a benefit over Euclid's method as I advance in Algebra or higher math?

Q3) My book shows the solution below to simply the fraction below.

3990=6* 665=665=7*95=95

6762 6*1127 1127 7*161 161

I don't understand how they arrived at the 6 and 7.

I did prime factorization and wrote out the following. (sorry, i used the , for multiplication on accident)

3990=2,3,7,5,19=(2,3,7), 5,19=95

6762 2,3,7,7,23 (2,3,7), 7,23 161

I notice I could multiply 2*3 and arrive at the 6 and 7 like they did, but then that's doing more steps in the method I'm doing.

So I'd like to know ...

Q4) Is there is a simpler method they used to arrive at the 6 and 7?

Also, a further note, using Euclid's Algorithm was one step faster than the prime factorization in this particular problem, plus I didn't have to refer to (or memorize) a table of prime factors.

Thanks for taking the time to read my post to the end. I hope it's clear and easy to understand.