How many four digit numbers $$\overline{abcd}$$, with non-zero digits $$𝑎, 𝑏, 𝑐, 𝑑$$ in base $$10$$, are there such that $$𝑎 + 𝑐 = 𝑏𝑑$$ and $$𝑏 + 𝑑 = 𝑎𝑐?$$

Let the required four-digit number be $$\overline{abcd}$$, where the digits $$a,b,c,d$$ take values from $$\{1,2,\dots ,9\}$$ (zero is not allowed).

The given conditions are
$$a+c = bd \qquad -(1)$$
$$b+d = ac \qquad -(2)$$

Because every digit is at most $$9$$, the largest possible value of the left-hand sides (the sums) is $$9+9 = 18$$. Hence from $$-(1)$$ and $$-(2)$$ we immediately get

$$bd \le 18, \qquad ac \le 18 \qquad -(3)$$

Thus only those ordered pairs of digits whose product does not exceed $$18$$ can appear in the pairs $$(b,d)$$ and $$(a,c)$$. This already eliminates the large majority of possibilities.

Fix an ordered pair $$(a,c)$$ that satisfies $$ac \le 18$$. Define
$$S_1 = a+c, \qquad S_2 = ac$$
Equations $$-(1)$$ and $$-(2)$$ can now be read as

$$bd = S_1, \qquad b+d = S_2$$

If we regard $$b$$ and $$d$$ as the two roots of a quadratic, they must satisfy

$$t^{2}-S_2\,t+S_1 = 0 \qquad -(4)$$

For $$b,d$$ to be (positive) integral digits, the quadratic in $$-(4)$$ must have

1. A non-negative discriminant: $$\Delta = S_2^{2}-4S_1 \ge 0$$.
2. A perfect-square discriminant (so that the roots are integral).
3. Both roots lying between $$1$$ and $$9$$ (inclusive).

We now list all ordered pairs $$(a,c)$$ with $$ac \le 18$$ and test them with the above three criteria. Because the list is short, the check can be done by hand in a few minutes; the successful pairs are summarised below.

Case 1: $$(a,c)=(1,5)$$
$$S_1 = 1+5 = 6,\; S_2 = 1\cdot5 = 5$$
$$\Delta = 5^{2}-4\cdot6 = 25-24 = 1 = 1^{2}$$
Roots $$b,d = \dfrac{5 \pm 1}{2} = 3,\,2$$ - both lie in $$\{1,\dots ,9\}$$.
This gives the two numbers $$1352$$ and $$1253$$. Case 2: $$(a,c)=(2,2)$$
$$S_1 = 2+2 = 4,\; S_2 = 2\cdot2 = 4$$
$$\Delta = 4^{2}-4\cdot4 = 0$$
Double root $$b=d=2$$, yielding the number $$2222$$. Case 3: $$(a,c)=(2,3)$$
$$S_1 = 2+3 = 5,\; S_2 = 6$$
$$\Delta = 6^{2}-4\cdot5 = 36-20 = 16 = 4^{2}$$
Roots $$b,d = \dfrac{6 \pm 4}{2} = 5,\,1$$.
Numbers obtained: $$2531$$ and $$2135$$. Case 4: $$(a,c)=(3,2)$$ (order reversed relative to Case 3)
$$S_1 = 3+2 = 5,\; S_2 = 6$$ - same discriminant and roots as before.
Numbers obtained: $$3521$$ and $$3125$$. Case 5: $$(a,c)=(5,1)$$ (order reversed relative to Case 1)
$$S_1 = 5+1 = 6,\; S_2 = 5$$ - same discriminant and roots as in Case 1.
Numbers obtained: $$5312$$ and $$5213$$.

No other ordered pair $$(a,c)$$ with $$ac \le 18$$ satisfies the discriminant and root conditions; hence the list above is complete.

Collecting all the valid four-digit numbers:

$$\{\,1253,1352,2135,2222,2531,3125,3521,5213,5312\,\}$$

The count of such numbers is $$9$$.

Answer: 09