Lucky 13

We require $a \in (1, 2, 3, 4, 5, 6, 7, 8, 9), b \in (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)$ when solving the quadratic equation in $a:$

$10a + b = a^2 + b^2 + ab \Rightarrow a^2 + (b-10)a + (b^2 - b) = 0$ (i)

which has roots: $a = \frac{(10-b) \pm \sqrt{(b^2 - 20b +100) - 4(1)(b^2 - b)}}{2} = \frac{(10-b) \pm \sqrt{100 - 16b - 3b^2}}{2}$ (ii).

In order for $a \in \mathbb{N}$ , we require a non-negative discriminant $\Rightarrow 100 - 16b - 3b^2 \ge 0$ , or:

$b = \frac{16 \pm \sqrt{256 - 4(-3)(100)}}{-6} = \frac{16 \pm 4\sqrt{91}}{-6} = \frac{8 \mp 2\sqrt{91}}{3} \Rightarrow -9.02 \le b \le 3.69$ ,

which limits us to just $\boxed{b = (0, 1, 2, 3)}.$ Testing each of these values in (ii) produces the ordered-pairs:

$(a,b) = (0,0); (10,0); (0,1); (9,1); (4+\sqrt{14}, 2); (4-\sqrt{14},2); (1,3); (6,3)$

of which only $\boxed{3}$ pairs work: $13, 63, 91$ .

Mathematica

Select[Range[10, 99],(s=IntegerDigits@#;s[[1]]^2+s[[2]]^2+s[[1]]*s[[2]]==#)&]

{13, 63, 91}