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## Homework Statement

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, 15q

^{2}= 5*3*q

^{2}= p

^{2}

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 15q

^{2}= 15*15*k

^{2}, so q

^{2}=15k

^{2}. 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?