"The remainders by the last digit."
Is 372 ÷ 4? Check last 2 digits: 72÷4=18 ✓ Yes!
Finding remainders and divisibility checks using digit patterns.
Use the last digit(s) to quickly determine remainders when dividing. Different divisors have different digit-checking rules.
This sutra leverages the fact that divisibility often depends only on the last few digits. Divisibility by 2: check last digit (even). By 4: check last two digits. By 8: check last three digits. By 5: last digit is 0 or 5. By 11: alternating sum of digits. These rules arise because powers of 10 have specific remainder patterns with each divisor. For example, 100 is divisible by 4, so only the last 2 digits matter for divisibility by 4. The sutra also covers finding actual remainders quickly: the remainder of 573 ÷ 4 is the remainder of 73 ÷ 4 = 1. This is invaluable for rapid mental checking of calculations.
Divisibility rules have been known in India since at least the Brahmasphutasiddhanta (628 CE), where Brahmagupta discusses divisibility by various numbers. The casting-out-nines method (related to this sutra) was used by Indian mathematicians long before it appeared in European texts. The Vedic system of checking answers using digit sums (a form of this sutra) was described extensively by Kenneth Williams in his Vedic Mathematics Teacher's Manual. The specific rules for divisibility by 7 and 11 were elaborated by Indian mathematicians like Mahavira (850 CE) and Narayana Pandit (1356 CE).
The divisibility rules are consequences of modular arithmetic. Since 10 ≡ 0 (mod 2, 5), only the last digit matters for divisibility by 2 or 5. Since 100 ≡ 0 (mod 4), only the last two digits matter for 4. Since 10 ≡ −1 (mod 11), the alternating-sum rule for 11 follows. These congruence relationships form the foundation of number theory. In cryptography, modular arithmetic is the basis of RSA encryption. In computer science, hash functions often use remainder operations for data distribution.