In the proof by contradiction of square root of two being irrational it is implied that if both a and b are even or odd then they cannot be on the lowest terms( $gdc(a, b) = 1$ ). Why must they be on the lowest terms for √2 to be rational?
√2 = a/b 2 = a²/b² 2b² = a² => 2 | a² => 2 | a 4 | a² 4 | 2b² 2 | b² => 2 | b
This proves that both a and b are even and thus have common factor of 2. Why does that contradict that √2 is rational?
One proof Why is it obvious
We additionally assume that this a/b is simplified to lowest terms, since that can obviously be done with any fraction.
Given the answers the question becomes why can we presume in the theorem that gdc(a, b) = 1 ?