Divisibility by primes

I n school we generally study divisibility by divisors from 2 to 12 (except 7 in some syllabi). In the case of composite divisors beyond 12, all we need to do is express the divisor as a product of coprime factors and then check divisibility by each of those factors. For example, take the case of 20; since 20 = 4 × 5 (note that 4 and 5 are coprime), it follows that a number is divisible by 20 if and only if it is divisible by 4 as well as 5. It is crucial that the factors are coprime. For example, though 10 × 2 = 20, since 10 and 2 are not coprime, it cannot be asserted that if a number is divisible by both 10 and 2, then it will be divisible by 20 as well. You should be able to find a counterexample to this statement. Divisibility tests by primes such as 7, 13, 17 and 19 are not generally discussed in the school curriculum. However, in Vedic Mathematics (also known by the name “High Speed Mathematics”; see Box 1), techniques for testing divisibility by such primes are discussed, but without giving any proofs. In this article, proofs of these techniques are discussed