Let T be the set of triangular numbers and T* be the set of all products of any two triangular numbers. Show that:

1. Among elements of T, each of the digits 0,1,5 and 6 occur in the units place twice as frequently as each of the digits 3 and 8. (More precisely, if MBk is the set of elements of T that are less than B and end in k, then, e.g., MB1/MB8 approaches 2 as B approaches infinity.)

2. None of the elements of T* end in 2 or 7.

The cycle of ending digits in multiplication of triangular numbers has a period of 20:
T_n {1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, 0}

As can be seen in the triangular numbers that only the digits 0, 1, 3, 5, 6 and 8 are in its ending-digit cycle.  The following shows the frequency of the digits in the cycle:

                       0-4; 1-4; 2-0; 3-2; 4-0
                       5-4; 6-4; 7-0; 8-2; 9-0

The digits 0, 1, 5, and 6 in the cycle occurs 4 times, which is twice as often as the digits 3 and 8. The digits 2, 4, 7, and 9 do not occur at all.

Looking at a multiplication matrix of the final digit of each of these digits one can see that 2 and 7 never appear.

   0 1 3 5 6 8 
0| 0 0 0 0 0 0 
1| 0 1 3 5 6 8 
3| 0 3 9 5 8 4 
5| 0 5 5 5 0 0 
6| 0 6 8 0 6 8 
8| 0 8 4 0 8 4 

Posted by Dej Mar on 2008-02-06 02:11:50