Persistence of a number

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In mathematics, the persistence of a number is the number of times one must apply a given operation to an integer before reaching a fixed point at which the operation no longer alters the number. The persistence of a number is undefined if a fixed point is never reached.

For example, the additive persistence counts how many times we must add its digits together to arrive at its digital root. The additive persistence of 2718 in base 10 is 2, as , and then that . Another example is the multiplicative persistence, which counts how many times we must multiply its digits together to arrive at its multiplicative digital root. The multiplicative persistence of 39 in base 10 is 3, because it takes three steps to reduce 39 to a single digit: .

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