Function Repository Resource:

# MultiplicativeDigitalRoot (1.0.0)current version: 1.1.0 »

Compute the multiplicative digital root of an integer

Contributed by: Christopher Stover
 ResourceFunction["MultiplicativeDigitalRoot"][n] returns the multiplicative digital root of n.

## Details

Starting with a non-negative integer n, multiply its digits, then multiply the digits of the resulting number, etc., until the result has only one digit. The one-digit integer marking the stopping point of this process is called the multiplicative digital root of n.
As an example, consider n=14691:
Multiplying its digits yields 1×4×6×9×1=216.
Multiplying the digits of the result yields 2×1×6=12.
Multiplying the digits of the result yields 1×2=2. Because 2 is a single-digit number, the process stops.
Because the integer marking the termination of the process is 2, ResourceFunction["MultiplicativeDigitalRoot"][14691] returns 2.

## Examples

### Basic Examples (1)

Verify the result claimed in the Details section:

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### Scope (2)

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Compute the multiplicative digital root of the first 100 integers (OEIS A031347):

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### Properties and Relations (4)

The result returned by MultiplicativeDigitalRoot can be manually computed using NestWhileList:

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The multiplicative digital root of an integer can also be manually computed with no high-level functions:

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Because testint3 is a single-digit integer, the process terminates here, and because testint3 equals 8, the multiplicative digital root of testint1=1234 is equal to 8:

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The number of iterations required to reach the end of the digit multiplication process is called the multiplicative persistence and is returned by the resource function MultiplicativePersistence:

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The additive analogue of the multiplicative digital root is called the additive digital root and is returned by the resource function AdditiveDigitalRoot:

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### Possible Issues (1)

MultiplicativeDigitalRoot requires its input to be non-negative:

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### Neat Examples (1)

Create a OEIS-themed table showing which integers have the same multiplicative digital root. This partially reproduces a cool result from the associated MathWorld article:

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therealcstover

## Version History

• 1.1.0 – 25 May 2023
• 1.0.0 – 09 August 2022

## Author Notes

Future versions of this will likely contain expansion to allow for integers in bases other than 10.
As mentioned on the page Problems & Puzzles, Puzzle 341, there is a modified definition of multiplicative persistence due to Erdős wherein one only multiplies the nonzero digits at each step. This modification is not implemented herein but may be included in a future update.
In a future update, I'm going to make the "Neat Examples" table look better. That's definitely a version 2 edit!