How do you find the last 3 non zero digits at the end of 3 to power 3333?

Answer 1

Before you embark on this calculations there are a few things you need to understand:

You are not going to be able to do it simply using a calculator or ordinary computer. The value of 33333 is simply too large - more than a googol15. With Excel, for example, you get a #NUM! error.

It is only the last 3-digits of any power of 3 that will contribute to the last 3-digits of the next power. All the earlier digits of the previous power of 3 can, therefore, be ignored in all subsequent calculations.

Last, since there can only be at most one thousand 3-digit endings (000 to 999), the numbers must start repeating if you go to the power 3333. And once they do, they will repeat the same sequence over and over again.

So, start with 30 = 1:

30 = 1

31 = 3*1 = 3

32 = 3*3 = 9

33 = 3*9 = 27

34 = 3*27 = 81

35 = 3*81 = 243

36 = 3*243 = 729

37 = 3*729 = 2187 but you only need the last 3 digits which are 187

Last three digits of 38 = last 3 digits of 3*187 = 561 which are 561 Last three digits of 39 = last 3 digits of 3*561 = 1683 which are 683

and so on

Last three digits of 3100 = last 3 digits of 3*667 = 2001 which are 001

So 3100 is equivalent (in the context of last 3 digits of powers of 3) to 30. That is to say, 3100 contributes 1 to the multiplication. Since multiplication by 1 can be ignored, all blocks of 3100 can be ignored. Therefore 33333 will be equivalent to 333.

And that gives 523.