Prove Square Root of 15 is Irrational
The Attempt at a Solution
Here's what I have. I believe it's valid, but I want confirmation.
As usual, for contradiction, assume 15.5=p/q, where p,q are coprime integers and q is non-zero.
Thus, 15q2 = 5*3*q2 = p2
Since 5 and 3 are prime, they must divide p. However, since the lcm(5,3) = 15, it must be the case that 15 divides p. Thus, p=15k for some k.
Then 15q2 = 15*15*k2, so q2=15k2. By the same argument, this implies 15 divides q.
However, we have reached a contradiction, since we assumed that p and q were coprime.
Is this a valid argument?