Problem 5

 

The digital root of a number is obtained by adding the digits of the number until a single digit is obtained.  The digital root of 943561 is 1 because 9 + 4 + 3 + 5 + 6 + 1 = 28, 2 + 8 =10, 1 + 0 =1. A number is called a triangular number, if it is equal to the sum of the first n natural numbers. Find all digital roots of triangular numbers that are not triangular numbers themselves.

 

[Problem submitted by Iris Magee, LACC Associate Professor of Mathematics.]

 

Solution for Problem 5:

 

The only such digital root is 9.  Observe in the following chart that the “digital-root pattern” repeats.  The only digital roots of triangular numbers are 1, 3, 6, and 9.  The first 3 are triangular numbers themselves.

 

Triangular Numbers

Digital Roots

1

1

3

3

6

6

10

1

15

6

21

3

28

1

36

9

45

9

55

1

66

3

78

6

91

1

105

6

120

3

136

1

153

9

171

9