Factorization of integers

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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.
 

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SteamKing
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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
 
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Stephen Tashi
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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?
 
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PeroK
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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.
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
 
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If you want to solve a quadratic equation by factorisation the you need to be able to factorises integers.
That is to solve

ax2 + bx + c = 0

you need to factorises a and c.
 
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