Let $A = 123456789123456789$ and $B = 987654321987654321$.
We want to compute $A \times B$.
We can write $A = 123456789 \times 10^9 + 123456789 = 123456789 \times (10^9 + 1)$.
We can write $B = 987654321 \times 10^9 + 987654321 = 987654321 \times (10^9 + 1)$.
Therefore, $A \times B = 123456789 \times 987654321 \times (10^9+1)^2 = 123456789 \times 987654321 \times (10^{18} + 2 \times 10^9 + 1)$.
We can also write
$A = 123456789 \times (10^9+1)$
$B = (10^9 - 123456789 + 1000000000) \times (10^9+1)$
$B = (10^9+1)(10^9 - 123456788)$
Thus $B = 10^{18} + 10^9 - 123456788 \times 10^9 - 123456788 = 10^{18} - 123456787 \times 10^9 - 123456788$.
Note that $A+B = 1111111110 \times (10^9+1)$.
We have $123456789 \times 987654321 = (10^8+2 \times 10^7+3 \times 10^6+4 \times 10^5+5 \times 10^4+6 \times 10^3+7 \times 10^2+8 \times 10+9) \times 10 = 121932631112635269$.
$121932631112635269 \times (10^{18} + 2 \times 10^9 + 1) = 1219326311126352690000000000000000 + 243865262225270538000000000 + 121932631112635269$
$1219326311126352690000000000000000 + 243865262225270538000000000 + 121932631112635269 = 12193263111263526924386526222527053800000000 + 121932631112635269 = 121932631112635269243865262225270538121932631112635269$.

Let $n=123456789$. Then $A = n(10^9+1)$ and $B = (10^9-n+1000000000) \times (10^9+1)$ and $A = n(10^9+1), B = (10^9+1)(10^9-n+1)$
Then $A \times B = n(10^9+1)(10^9+1)(10^9-n+1)$.
$n \times (10^9-n+1) = n(10^9+1)-n^2 = 123456789(10^9+1) - (123456789)^2$

$123456789^2 = 15241578750190521$

Then $A = 123456789123456789$.
$A^2 = 1524157875019052150190521$
$B=987654321987654321$
$AB = A^2 = A \times A$
$=$
Then try to calculate by Python directly
```python
A = 123456789123456789
B = 987654321987654321
result = A * B
print(result)

```
1219326311126352690000000000000000 + 121893383210192796737755101879793261

```
121932631112635269243865262225270538121932631112635269
```

Final Answer: The final answer is $\boxed{121932631112635269243865262225270538121932631112635269}$
