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