Question

* Can every composite numeber be written as a product of primes? Explain your reasoning

* If two people write the same number as a product of prime: How would their factorizations be alike? How might the factorization be different?

Answer 1

1. Yes, every composite number can be written as product of primes. By definition any positive number is either prime or composite except 1. If the number is prime, then its only divisors are 1 and itself. However, if the number is composite we will have divisors which are different from 1 and itself.

Let's assume that for all number till n for n greater than 1, the given statement is true. For n = 2, we know that 2 is prime, hence initial case is true. Now consider n+1. Either n+1 is prime, hence we stop or n+1 can be written as product of 2 numbers neither of which is 1. Those smaller integers are either primes or product of primes so n being a product of product of primes is also a product of primes.

2. If two people write the same number as a product of primes, their factorization would always be alike and will exactly contain the same prime numbers with same frequency. For example 100 will always be factorised as 2*2*5*5. It will always have two 2s and two 5s. Only the order of factorization may be different. For example one may express it as 2*5*2*5 while the other may express it as 5*2*2*5