Ekadhikena Purvena

Meaning: "By one more than the previous one". This sutra simplifies squaring numbers ending in 5 and certain multiplications where the last digits sum to 10 and preceding digits are the same.

Why is it super useful?

Squaring Numbers Ending in 5: If you need to find the square of any number that ends with the digit 5 (like 15, 25, 35, 105, etc.), this sutra makes it super easy and fast! No long multiplication needed. The process is:

1. The last part of the answer is always 25 (since 5x5=25).
2. For the first part, take the digit(s) before the 5, let's call this 'N'. Multiply N by (N+1). Combine this result with 25 to get your answer.

Special Multiplications: It can also be used for some other specific types of multiplication, especially when numbers are close to multiples of 10, or when one number can be conveniently related to 'one more than' a part of the other. For instance, multiplying two numbers where the sum of their last digits is 10 and the preceding digits are the same (e.g., 43 x 47). Here, the last part is 3x7=21. The first part is 4 x (4+1) = 4x5 = 20. So, 43x47 = 2021.

Are there any rules or restrictions?

For Squaring: The main restriction (and the magic part!) is that the number must end in 5. It won't work directly for squaring numbers like 23 or 47 if you're trying to use the "ending in 5" shortcut.

For General Multiplication (sum of last digits is 10): The preceding digits of both numbers must be identical. For example, it works for 62 x 68 but not for 62 x 78.

Conceptual Understanding: This sutra is a specific application of algebraic identities. For numbers ending in 5, like (10N+5)², it expands to 100N² + 100N + 25, which is 100N(N+1) + 25. This directly shows why we multiply N by (N+1) for the first part and append 25.