Last Digits

We often approximate the numbers for convenience and the digits to the rightmost side are neglected the most. I wanted to see of there is any pattern in the ignored right digits. For this problem, I have considered the increasing powers of a number. For example, have a look at this sequence.

21 = 2
22 = 4
23 = 8
24 = 16
25 = 32
26 = 64
27 = 128

Have you found any pattern ?
The last digits are repeating as 2,4,8,6,2,4,8.
What if this is done not just for 2 but for the next numbers too ?
Let us see it first before plotting.

31 = 3
32 = 9
33 = 27
34 = 81
35 = 243
36 = 729
37 = 2187

It is more apparent now isn’t it. The sequence is 3,9,7,1,3,9,7………

This being the case, I have taken it a few more steps ahead. I have run the same sequence with last two digits, three digits, four digits and so on. After proceeding to the higher numbers, I was taken aback by the visual patters that were emerging from the seemingly mundane and neglected string of integers.

For plotting, what has been done is Cn for increasing n (which means increasing powers of a constant number)

2n for last two digits

2n for last three digits



2n for last four digits


I have skipped the other one digit numbers for convenience but there were similar patterns in them too. The most intriguing ones are shown below.

last 5 digits of  11n


last 6 digits of  11n


last 5 digits of  13n


last 6 digits of  14n


last 7 digits of 14n


last 6 digits of 16n


last 5 digits of 18n


last 6 digits of 18np1806u

last 4 digits of 19n


last 5 digits of 19n


last 4 digits of 21n


last 5 digits of 21n


This post will be updated when other similar patterns are discovered.
















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