Why is factorization of integers important on a first number theory course? Where is factorization used in real life? Are there examples which have a real impact? I am looking for examples which will motivate students.
I'll take a stab. Factorization helps determine if a given integer is prime, and one use for prime integers is in devising cryptography keys, which are used quite a bit for, among other things, encrypting sensitive data which might be swapped around on the internet. (NSA, how'm I doin' so far?) If you have an arbitrary integer of n-digits, how long does it take to determine the factors (if any) of this integer? That's a pretty basic question for number theory to answer. Is it a couple of hours, a couple of days, a couple of years, a couple of centuries, or what? Can a better (= quicker) algorithm be devised? http://en.wikipedia.org/wiki/Factorization
Suppose we can motivate an interest in Diophantine equations. Their solution entails finding greatest common divisors. Would that also lead in a natural way to focusing on prime numbers?
Much of Internet security uses Public Key Cryptography, which depends on large integer factorisation. See, for example: http://en.wikipedia.org/wiki/Public-key_cryptography
If you want to solve a quadratic equation by factorisation the you need to be able to factorises integers. That is to solve ax^{2} + bx + c = 0 you need to factorises a and c.