Technique explanation
This special case multiplication applies when the leading digits of two numbers are identical and their trailing digits (tails) sum to a power of ten (10, 100, 1000, etc.). The product is found in two parts: the left part is the leading part multiplied by 'one more than itself', and the right part is the product of the tails.
Method Condition: Leading digits must be identical; trailing digits must sum to a power of ten.
Standard fallback: Standard long multiplication or the distributive property (e.g., 47 × 40 + 47 × 3).
Worked examples
Example 1: Base 10 Tail
Calculate 47 × 43.
- Check condition: Leading parts are both 4. Tails 7 + 3 = 10.
- Left part: 4 × (4 + 1) = 20.
- Right part: 7 × 3 = 21.
- Result: 2021.
- Check: 47 × 43 = (45+2)(45-2) = 45² - 4 = 2025 - 4 = 2021 (Difference of squares).
Example 2: Base 100 Tail
Calculate 792 × 708.
- Check condition: Leading parts are both 7. Tails 92 + 08 = 100.
- Left part: 7 × 8 = 56.
- Right part: 92 × 08 = 736. Reserve 4 digits (2 × tail width) on the right: 0736.
- Result: 560736.
- Check: 792 × 708 = 560736 (Verified by standard long multiplication).
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