TITLE: Why do we consider prime numbers  important, and what are their applications other than number theory in pure math?
QUESTION [11 upvotes]: Why do we consider prime numbers  important, and what are their applications other than number theory in pure math? I know that Number theory is devoted to studying prime numbers, but there must be other reason to study the properties of prime numbers, right?

REPLY [5 votes]: Relative primes are also useful when building gear trains. It helps to reduce wear. If one gear has a number of teeth that is a factor of the other gear the same teeth always come into contact with one another. 
For instance if one gear has 10 teeth and the other 20, the first tooth on the 10t gear always comes into contact with the 1st and 11th teeth on the 20t gear. 
If, on the other hand, one gear has 11 teeth and the second has 20 teeth (20 is not prime but 11 and 20 have no common factors) the first rotation the first tooth contacts the 1st and 12th teeth on the first revolution and the 3rd and the 14th on the second, the 5th and 16th on the third, and it will take 21 revolutions to get back to the first tooth.
In nature prime numbers show up time to time. Cicadas live under ground for prime numbers of year before coming up and making lots of racket (here is a blog post about it http://www.factodiem.com/2010/04/prime-years-of-life.html)
Of course lots of thing are important to study, not because of what they HAVE been used for but for the things they WILL be used for.
Hope this helps.