"By one less than the previous."
76×99 → Left: 76-1=75, Right: 99-75=24 → 7524
Multiplying any number by 9, 99, 999, etc. Incredibly fast!
When multiplying by numbers made entirely of 9s, subtract 1 from the multiplicand for the left part, then find the complement for the right part.
This sutra provides lightning-fast multiplication by 9, 99, 999, etc. The method: Left part = number − 1. Right part = complement of left part (from 9s). For 76 × 99: Left = 76 − 1 = 75. Right = 99 − 75 = 24. Answer: 7524. Why it works: 76 × 99 = 76 × (100 − 1) = 7600 − 76 = 7524. The "one less" gives the thousands/hundreds, and the complement gives the remainder. For single-digit × 9: 7 × 9 → left: 7−1=6, right: 9−6=3, answer: 63. The 9-times table pattern: left digit goes up (1,2,3...) while right digit goes down (9,8,7...), and digits always sum to 9.
The 9-times pattern was recognized in ancient India and encoded in finger-counting methods still taught in Indian schools. The technique of "multiplying by one less than a power of 10" directly connects to Sutra 2 (Nikhilam) — both exploit the complement from powers of 10. Medieval Indian merchants used this technique extensively for pricing goods in nines (a marketing strategy that predates modern "psychological pricing" by centuries). The mathematical trick of n × 9 = 10n − n was formalized in the Vedic system as this specific sutra, providing a memorable rule for all powers of 9.
The proof is simple: n × (10^k − 1) = n × 10^k − n. The left part is n−1 (since the subtraction borrows from the highest digit), and the right part is the complement. This connects to the mathematical concept of the "repunit" (111...1) and its relationship to cyclic numbers. In number theory, multiplying by 9 and checking digit sums is a fundamental primality test component. In computing, multiplying by (2^n − 1) is used in hash functions and pseudo-random number generators.