Prove the following statemnet: any natural number greater than 1 is divisible by a prime number

See the proof below

Step-by-step explanation:

Proof

We can proof this using the well ordering principle.

Let's assume that k is a prime number >1. If k is a prime by definition is divisible by the prime number k.

Let's assume now that k is composite with the following form where represent integers less than the number k.

Assuming that is prime then we have that and we satisfy the conditions.

Now let's assume that is also composite with the following form as the product of two integers and for we assume that both are integers >1 and smaller than

If we continue this process t times dividing the composite factors into products of smaller factors we satisfy the condition that the set {} would be non empty.

And using the Well ordering principle we have t elements. From this we can conclude that is prime and if we discompose this number we satisfy the existance of less than because we have this:

with k a divisible prime.

And with this we satisfy the conditions and then the proof is complete.  

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step-by-step explanation: