Units Digit of 2^57

Date: 10/07/2001 at 03:11:52
From: Lauren Wyluda
Subject: 8th grade algebra

What is the units digit of 2 to the power of 57? 

This would help me out tremendously. Thank you,  
Lauren

Date: 10/07/2001 at 11:29:27
From: Doctor Paul
Subject: Re: 8th grade algebra

Look at the units digits of the following powers of 2:

 2^1 = 2
 2^2 = 4
 2^3 = 8
 2^4 = 16
 2^5 = 32
 2^6 = 64
 2^7 = 128
 2^8 = 256
 2^9 = 512
2^10 = 1024

Do you see a pattern?  The units digits cycle through the numbers

2, 4, 8, 6

and then start over again.

So when n is a multiple of four, 
the units digit of 2^n is 6.

When n is one more than a multiple of four, 
the units digit of 2^n is 2.

When n is two more than a multiple of four, 
the units digit of 2^n is 4.

When n is three more than a multiple of four, 
the units digit of 2^n is 8.

Since 57 = 14*4 + 1 is one more than a multiple of four, it follows 
that the units digit of 2^57 is 2.

In fact, 2^57 = 144115188075855872 and so the formula works.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/