Question

a) State the well-ordering property for the set of positive integers.

b) Use this property to show that every positive integer greater than one can be written as the product of primes.

Short answer

b)\({n^*} = {p_1},\_\_\_\_,{p_k}.{q_1},\_\_\_\_,{q_m}\).

Step-by-step solution

a) State the well-ordering positive integers.

Every nonempty set of nonnegative integers has the smallest element.

In any set of positive integers is the smallest number.

b) show that every positive integer greater than one can be written as the product of primes.

1) Explanation

Assume there exists some positive integer

that cannot be written as the product of primes. Let S be the set of positive integers that do not have a prime factorization.

There is a smallest positive integer that cannot be written as the product of primes, this is\({n^*}\).

2) \({n^*}\) is prime.

Here the prime factorization is \({n^*}\)itself.

But here got a contradiction.

\({n^*} \in S\)so by definition does not have a prime factorization.

3) \({n^*}\)is not prime.

Let \({n^*} = ab\)

Here\(a \ne {n^*}\), \(a \ne 1\)and\(b \ne {n^*}\), \(b \ne 1\).

This implies that \(a < {n^*}\)and\(b < {n^*}\).

Here \({n^*}\)are the smallest elements of\(S\), \(a \notin S\) and\(b \notin S\).

4) Solution

Therefore \(a,b\)prime factorizations.

\(\begin{array}{c}a = {p_1},\_\_\_\_,{p_k}\\b = {q_1},\_\_\_\_,{q_m}\end{array}\)

Prime number \({p_1},\_\_\_\_,{p_k},{q_1},\_\_\_\_,{q_m}\)but,

\(\begin{array}{c}{n^*} = a.b\\ = {p_1},\_\_\_\_,{p_k}.{q_1},\_\_\_\_,{q_m}\end{array}\)

This is a prime factorization\({n^*}\).