1. Let $a$ and $b$ be rational numbers such that $s = a + b = a^2 + b^2$. We need to prove that $s$ can be written as a fraction where the denominator is relatively prime to $6$.

2. First, consider the case where $ab = 0$. If either $a$ or $b$ is zero, then:
- If $a = 0$, then $s = b = b^2$. This implies $b = 0$ or $b = 1$, so $s = 0$ or $s = 1$.
- If $b = 0$, then $s = a = a^2$. This implies $a = 0$ or $a = 1$, so $s = 0$ or $s = 1$.

In both cases, $s$ is either $0$ or $1$, which are integers and can be written as fractions with denominators $1$, which is relatively prime to $6$.

3. Now, consider the case where $ab \neq 0$. Let $a = \frac{u}{v}$ and $b = \frac{w}{t}$, where $u, v, w, t \neq 0$. Then:
$$s = \frac{u}{v} + \frac{w}{t}$$
Multiplying both sides by $vt$, we get:
$$svt = ut + vw$$
Squaring both sides, we obtain:
$$s^2 v^2 t^2 = (ut + vw)^2$$
Let $m = ut$ and $n = vw$. Then:
$$s^2 v^2 t^2 = (m + n)^2$$
Also, we have:
$$s = \frac{u^2}{v^2} + \frac{w^2}{t^2}$$
Multiplying both sides by $v^2 t^2$, we get:
$$s v^2 t^2 = u^2 t^2 + w^2 v^2$$
Thus:
$$s = \frac{u^2 t^2 + w^2 v^2}{v^2 t^2}$$
Let $m = ut$ and $n = vw$. Then:
$$s = \frac{m^2 + n^2}{v^2 t^2}$$

4. Consider the parity of $m$ and $n$:
- If $m^2 + n^2$ is even, then either both $m$ and $n$ are even, or both are odd.
- If both $m$ and $n$ are even, we can replace $m$ and $n$ with $\frac{m}{2}$ and $\frac{n}{2}$, respectively.
- If both $m$ and $n$ are odd, then $m^2 + n^2 \equiv 2 \pmod{4}$ and $(m + n)^2 \equiv 0 \pmod{4}$. We can divide both parts by $2$ and get a fraction with an odd denominator.

5. Consider divisibility by $3$:
- If $m^2 + n^2 \equiv 0 \pmod{3}$, then both $m$ and $n$ are divisible by $3$. We can replace $m$ and $n$ with $\frac{m}{3}$ and $\frac{n}{3}$, respectively.

6. By repeatedly applying the above steps, we can always write $s$ as a fraction whose denominator is odd and not divisible by $3$. Therefore, the denominator is relatively prime to $6$.

$\blacksquare$