gmatutor wrote:Of course for the GMAT all primes are positive.
However, negative one satisfies the Fundamental Theorem and the most common definition of a prime.
Positive one is not considered prime because it is divisible be exactly one integer rather than two.
Negative one on the other hand is divisible by exactly two integers (+1 and -1). Thus negative one satisfies the definition of a prime.
The most common definition of a prime is the following: 'primes are positive integers which are divisible by precisely two distinct positive integers.' If you remove the word 'positive' from the definition, the only primes would be 1 and -1 (which both have two integer divisors, 1 and -1). 3 would not be prime- it is divisible by 1, -1, 3 and -3.
gmatutor wrote:
Since that is the case, the Fundamental Theorem is satisfied if we do not let the prime factor negative one have a power other than one. In other words we cannot factor 5 into 5 X -1^2.
Yes, but the restriction you need to impose on the powers of -1 here are similar to the restrictions as you would need to impose on the power on 1 in a prime factorization if you permitted 1 to be prime. Mathematicians do not permit 1 to be prime for this very reason; if 1 is not prime, the Fundamental Theorem of Arithmetic can be stated without any restrictions on powers.
