Special Case: Same Tens Digit, Units Add to 10

The Beautiful Special Case

When two 2-digit numbers have:

• Same tens digit
• Units digits that add to 10

There's an instant method!

Examples of This Pattern

• 23 × 27 (both have tens=2, and 3+7=10)
• 41 × 49 (both have tens=4, and 1+9=10)
• 65 × 65 (both have tens=6, and 5+5=10)
• 82 × 88 (both have tens=8, and 2+8=10)

The Formula

For numbers like n3 × n7 where n is the tens digit:

1. Left part: n × (n+1)
2. Right part: 3 × 7 = 21
3. Answer: Concatenate them

Example: 23 × 27

• Left: 2 × (2+1) = 2 × 3 = 6
• Right: 3 × 7 = 21
• Answer: 621

Example: 41 × 49

• Left: 4 × (4+1) = 4 × 5 = 20
• Right: 1 × 9 = 09 (pad to 2 digits)
• Answer: 2009

Why It Works

For (10n + a) × (10n + b) where a + b = 10:

= 100n² + 10n(a+b) + ab
= 100n² + 10n(10) + ab
= 100n² + 100n + ab
= 100n(n+1) + ab

So: n(n+1) | ab

Special Note: This Includes Squaring

Numbers ending in 5 are a special case:

• 25 × 25: tens=2, units 5+5=10 ✓
• 65 × 65: tens=6, units 5+5=10 ✓

This is why squaring numbers ending in 5 works:

• n5² = n(n+1)|25

Pattern in Action

11 × 19 = 1×2 | 1×9 = 2|09 = 209
22 × 28 = 2×3 | 2×8 = 6|16 = 616
33 × 37 = 3×4 | 3×7 = 12|21 = 1221
44 × 46 = 4×5 | 4×6 = 20|24 = 2024
55 × 55 = 5×6 | 5×5 = 30|25 = 3025

Notice the elegant progression!