Prove that if p is a prime number,

Wade Bullock

Wade Bullock

Answered question


Prove that if p is a prime number, then p is an irrational number.

Answer & Explanation

Elijah Benjamin

Elijah Benjamin

Beginner2022-07-16Added 10 answers

By way of contradiction, assume p is rational. Then there exist a , b Z with b 0 such that p = a b . Without loss of generality, we may assume gcd ( a , b ) 1
We can make this assumption, because we still lose no generality.
Now using gcd ( a , b ) = d 1. Then we can write a = d a and b = d b , for some relatively prime integers a and b .
p = a b = d a d b = a b ,
So we have shown that p is a ratio of two relatively prime integers.

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