# Multiplying 123456789123456789 × 987654321987654321

Let me compute this step by step using the structure of these numbers.

**Setting up the problem:**
- Let A = 123456789123456789 = 123456789 × (10⁹ + 1)
- Let B = 987654321987654321 = 987654321 × (10⁹ + 1)

So A × B = 123456789 × 987654321 × (10⁹ + 1)²

**Step 1:** Compute 123456789 × 987654321

123456789 × 987654321 = **121932631112635269**

**Step 2:** Compute (10⁹ + 1)² = 10¹⁸ + 2×10⁹ + 1

**Step 3:** Multiply 121932631112635269 by (10¹⁸ + 2×10⁹ + 1)

This means adding three shifted copies:
```
  121932631112635269000000000000000000   (× 10¹⁸)
+        243865262225270538000000000     (× 2×10⁹)
+                 121932631112635269     (× 1)
```

Aligning and adding:

```
121932631112635269000000000000000000
        243865262225270538000000000
                 121932631112635269
─────────────────────────────────────
121932631112879134262225391635269
```

**Final Answer:**

$$123456789123456789 \times 987654321987654321 = \boxed{121932631112879134262225391635269}$$
