# Multiplying 123456789123456789 × 987654321987654321

I need to compute this product exactly. Let me notice the structure of these numbers.

Let me define:
- A = 123456789
- B = 987654321

So the two numbers are:
- First number: A × 10⁹ + A = A × (10⁹ + 1)
- Second number: B × 10⁹ + B = B × (10⁹ + 1)

So the product is: **A × B × (10⁹ + 1)²**

## Step 1: Compute A × B = 123456789 × 987654321

I'll use the fact that 987654321 = 10⁹ - 12345679.

So A × B = 123456789 × (1000000000 - 12345679)

- 123456789 × 1000000000 = 123456789000000000
- 123456789 × 12345679

For 123456789 × 12345679:
Note that 12345679 × 9 = 111111111, so 12345679 = 111111111/9

123456789 × 12345679 = 123456789 × 111111111 / 9

123456789 / 9 = 13717421 (since 9 × 13717421 = 123456789)

So 13717421 × 111111111 = 1524157875308631

Therefore: A × B = 123456789000000000 - 1524157875308631 = **121932631124691369**

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

## Step 3: Compute A × B × (10⁹ + 1)²

= 121932631124691369 × (10¹⁸ + 2000000002 × 10⁹ + 1)

Wait, let me reconsider: (10⁹ + 1)² = 10¹⁸ + 2·10⁹ + 1

So I need:
121932631124691369 × 10¹⁸ = 121932631124691369 000000000000000000

Plus 121932631124691369 × 2000000000:
= 243865262249382738 000000000

Plus 121932631124691369 × 1:
= 121932631124691369

Wait, 2 × 10⁹ = 2000000000, so:

121932631124691369 × 2000000000 = 243865262249382738000000000

Now adding:
```
  121932631124691369 000000000000000000
+            243865262249382738 000000000
+                    121932631124691369
```

Let me align by digits (the first number has 36 digits):
```
  121932631124691369 000000000 000000000
  000000000243865262 249382738 000000000
  000000000000000000 121932631 124691369
```

Adding column by column (in 9-digit blocks from right):
- Rightmost 9: 000000000 + 000000000 + 124691369 = 124691369
- Next 9: 000000000 + 249382738 + 121932631 = 371315369
- Next 9: 000000000 + 243865262 + 000000000 = 243865262 (but need to check carry — no carry from 371315369)
- Wait, 249382738 + 121932631 = 371315369 ✓ (no carry)
- Next 9: 121932631 + 000000000 = 121932631... but we need to add the upper part of 243865262:

Let me regroup into 9-digit blocks:
```
  121932631 124691369 000000000 000000000
  000000000 243865262 249382738 000000000
  000000000 000000000 121932631 124691369
```

Adding:
- Block 1 (rightmost): 000000000 + 000000000 + 124691369 = 124691369
- Block 2: 000000000 + 249382738 + 121932631 = 371315369
- Block 3: 124691369 + 243865262 + 000000000 = 368556631
- Block 4: 121932631 + 000000000 + 000000000 = 121932631

## Final Answer

$$\boxed{121932631368556631371315369124691369}$$
