We're talking about primes and the unique factorization theorem in my applied algebra class. Something I'm not understanding is prime factorization. For example, say we want gcd(21000, 4400).
To start off, my professor showed that 21,000=2^(3) x 3^(1) x 5^(3) x 7^(1) and that 4400=2^(4) x 5^(2) x 11^(1).
I don't understand how in the heck you figure that out. Do you just have to sit there a long time and play with numbers until you finally get something that works or is there some kind of method for figuring out what primes to use and what the exponents on those primes are? Like, how do you randomly know to use the primes 2,3,5, and 7 for 21,000 and how do you automatically know what the exponent on them should be? Is there something I'm missing?