A General Rule for Divisibility Based on the Decimal Expansion of the Reciprocal of the Divisor

Abstract

1. The following rule provides a general test for determining the divisibility of an integer N= ,I0 ai 10i by any integer n: 1. Find p and q, where p is the number of terms preceding the period in the decimal expansion of l/n and q is the number of terms in the period. They can be found directly or by applying the theorem which states that, if n=no2a5f where no and 10 are relatively prime, then p =max (a, i3) and q is the exponent to which 10 belongs modulo no. (The number of digits in N must exceed p+q for the rule to have any significance.) 2. Mark off the digits of N in groups of q beginning with the (p l)th digit from the right. (Up to q -1 zeros may have to be placed before N to fill out the last group; this gives [(k-p)/q] + 1 groups.) 3. Find the sum, S, of the q-digit numbers formed in 2. 4. Substitute S for the digits of N preceding the last p digits, obtaining the smaller number N'. The rule states that if N' is divisible by n, so is N. Proof. Since