Prove that if p is a prime number,

Wade Bullock 2022-07-15 Answered
Prove that if p is a prime number, then p is an irrational number.
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Answers (1)

Elijah Benjamin
Answered 2022-07-16 Author has 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 .
Hence
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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