Persistence of a number assumed. The single-digit final state reached in the process of calculating an integer's additive persistence is its digital root. Put another way, a number's additive persistence counts how many times we must sum its digits to arrive at its digital root. The additive persistence of 2718 is 2: first we find that 2 + 7 + 1 + 8 = 18, and then that 1 + 8 = 9. The multiplicative persistence of 39 is 3, because it takes three steps to reduce 39 to a single digit: 39 → 27 → 14 → 4. Also, 39 is the smallest
International Standard Book Number that the sum of all the ten digits, each multiplied by its weight in "ascending" order from 1 to 10, is a multiple of 11. For this example: Formally, we can say: The two most common errors in handling an ISBN (e.g., typing or writing it) are a single altered digit or the transposition of adjacent digits. It can be proved that all possible valid <nowiki>ISBN-10</nowiki>'s have at least two digits different from each other. It can also be proved that there are no pairs of valid <nowiki>ISBN-10</nowiki>'s with eight identical digits and two transposed digits. (These are true only