In any primer on Number Theory, a primary factor is the study of prime numbers. (Sorry, I couldn't resist that, um, pun-filled line.)There are many interesting relationships that arise from the properties of primes (an integer divisible only by 1 and itself). Just for fun, I'll be posting occasional factoids involving primes or basic number theory. Even if you're not a math geek, I promise your head won't explode.
First up: Sphenic Numbers
A sphenic number is an integer that results from multiplying 3 distinct primes. For example, the first sphenic number is 30, which you get by multiplying the primes 2, 3, and 5.
n = p1 * p2 * p3 where pn are primes
30 = 2 * 3 * 5
The first 10 sphenic numbers are 30, 42, 66, 70, 78, 102, 105, 110, 114, and 130. Note that 105 is the very first odd sphenic number and shares a few other notable properties.
Every sphenic number has exactly 8 divisors, no more and no less. For 30, the divisors are {1, 2, 3, 5, 6, 10, 15, 30}; for 42 we get {1, 2, 3, 6, 7, 14, 21, 42}. While this might seem unusual at first glance, it's easy to demonstrate. Remember that every sphenic number is the product of 3 primes:
n = p * q * r
Since no prime has any factor other than itself (and 1), the only possible divisors of the result are:
{1, p, q, r, p*q, p*r, q*r, n}
Exercise: With any product of 4 unique primes, how many possible divisors will there be?
(Answer: Mouse over the image "105" in this blog) Post from : http://www.markvanorden.com/blog/mvoblog.php
Printed from : http://www.markvanorden.com/blog/mvoblog.php?id=18