Factors and PrimesDate: 09/10/97 at 12:13:07 From: Jacob Dick Subject: Factors and primes I'm student teaching right now in a middle school and I know that a prime number is a number greater than 1, and has only 2 divisors, 1 and itself. I was just curious... if they didn't make the stipulation greater than one, how much would that affect it? Wouldn't the only added number be negative one, since its factors would be 1 and -1? Date: 09/10/97 at 12:26:48 From: Doctor Rob Subject: Re: Factors and primes The Fundamental Theorem of Arithmetic, that every positive integer can be uniquely factored into a product of powers of prime numbers, would fail, since 3 = 1*3 = 1^2*3 = 1^3*3 = ... . It would have to be restated to say that every positive integer can be uniquely factored into a product of powers of prime numbers different from 1. Many other theorems would have to be restated to say things about "prime numbers different than 1." Example: every prime number (different from 1) has exactly two positive divisors. It is just simpler to exclude 1 from the set of prime numbers. Furthermore, 1 is a member of a different class: the units. They are integers whose reciprocal is also an integer. They consist of 1 and -1 only. The set of integers (or, more generally, any ring) is generally split into four sets: the zero-divisors, the units, the primes, and the composites. These are mutually exclusive and collectively exhaustive. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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