What is wrong with my proof by contradiction?

Solution 1:

I'm assuming you're in a number theory class or abstract algebra. At this level of such a course, we haven't formally reintroduced $\Bbb Q $ so fractions don't formally exist yet.

We have multiplication and addition. And we have the integers.

The better proof is to show that $\gcd(18,6)=6$ and hence that the smallest positive linear combination of $18$ and $6$ we can make is $6$.

Solution 2:

The core idea of your argument looks fine to me. I suspect what your professor had in mind, is that you should not write $3a + b = \frac{1}{6}$. You should simply say that $$ 6(3a + b)=1 $$ is not possible in integers, since $6$ is not an invertible element in $\mathbb{Z}.$

Solution 3:

$18a+6b=2\cdot(9a+3b)$ is an even number, so it cannot equal $1$ which is an odd number.