Multiplicative digital root

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The multiplicative digital root of a positive integer n is found by multiplying the digits of n together, then repeating this operation until only a single digit remains. This single-digit number is called the multiplicative digital root of n.[1]

Multiplicative digital roots depend upon the base in which n is written. If the term is used without qualification, it is assumed that n is written in base 10.

Multiplicative digital roots are the multiplicative equivalent of digital roots.


9876 would be reduced as 9876 -> 9×8×7×6 = 3024 -> 3×0×2×4 = 0. So the multiplicative digital root of 9876 is 0 and its multiplicative persistence (the number of steps required to reach a single digit) is 2.


  1. ^ Weisstein, Eric W. "Multiplicative Persistence". MathWorld.