examples of digital roots in a few selected bases


In base 10, the digital roots of the first few integers are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, … (listed in A010888 of Sloane’s OEIS), exhibiting a period of 9. Thus, all multiplesMathworldPlanetmathPlanetmath of 9 have digital root 9 in base 10 (in fact, this is the divisibility ruleMathworldPlanetmath for 9).

A prime in base 10 can have a digital root of 1, 2, 3, 4, 5, 7 or 8, or put another way, it can’t have a digital root of 6 or 9. The exclusion of 9 is not surprising, given that it is b-1. The inclusion of 3 is actually a special case: the only prime with that digital root is 3 itself. All others with digital root of 3 are multiples of 3, and the same goes for 6.

A square can have a digital root of 1, 4, 7 or 9, while a cube can have a digital root of 1, 8 or 9. Other figurate numbersMathworldPlanetmath show similar patterns in relation to their digital roots.

All factorialsMathworldPlanetmath for n>5 have digital root 9, since from that point forward they are all divisible by 9.

All perfect numbers have a digital root 1 in base 10, with the exception of 6. At least this is the case for the first 39 perfect numbers. It’s possible that if an odd perfect number exists it might have a different digital root.

A similar situation seems to hold in base 4, in which 6 has digital root 3 and the next ten or so all have a digital root of 1. They show more diversity in octal and hexadecimal.

In binary and factorial base, all positive integers have a digital root of 1. So, in a trivial sense, the digital root is a multiplicative functionMathworldPlanetmath in those bases.

Title examples of digital roots in a few selected bases
Canonical name ExamplesOfDigitalRootsInAFewSelectedBases
Date of creation 2013-03-22 16:00:10
Last modified on 2013-03-22 16:00:10
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 6
Author PrimeFan (13766)
Entry type Example
Classification msc 11A63