SEARCH HOME
 Math Central Quandaries & Queries
 Question from ronni, a student: why is the number 1 neither prime nor composite?

Hi Ronni,

When we are discussing primes and composites we are restricting ourselves to positive integers, so our discussions do not include negatives or zero.

I think it is easier to see why 1 is not a composite. The idea of a composite number is one that is built from other numbers. (Here the building tool is multiplication.) Thus 12 is composite since it can be built as 2 × 6 or even 3 × 4. I don't want to include 3 as a composite number even though it is 3 × 1 so I want my definition of composite to exclude 3. One way to accomplish this is to say a composite number is one that has more than 2 divisors. Thus 12 is composite, its divisors are 1, 2, 3, 4, 6 and 12 but 3 is not composite since its only divisors are 1 and 3. 1 also is not composite since its only divisor is 1.

So then should we say 1 ia a prime? Primes are the fundamental building blocks if again we are talking about multiplication as the building tool. Any number can be built as a product of primes. For example 60 = 2 × 2 × 3 × 5 = 22 × 3 × 5 and 2, 3 and 5 are primes. In fact as long as we agree that the order of multiplication is unimportant (that is for example 2 × 2 × 3 × 5, 2 × 3 × 2 × 5 and 2 × 5 × 3 × 2 are the same construction of 60) then any number can be built as a product of primes in only one way. Here I am thinking of 60 as built using multiplication from 2 twos, 1 three and 1 five. This is a very useful result. It is usually stated Every positive integer greater than 1 can be written uniquely as the product of primes. If 1 is a prime then this result is not true! 60 = 22 × 3 × 5 = 1 × 22 × 3 × 5 = 12 × 22 × 3 × 5 = 13 × 22 × 3 × 5 etc. This result that every positive integer greater than 1 can be written uniquely as the product of primes is so useful that mathematicians decided not to include 1 as a prime.

I hope this helps,
Harley

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.