Results 1 to 3 of 3

Math Help - What is the last digit of 2^1000

  1. #1
    Junior Member
    Joined
    Oct 2007
    Posts
    37
    Awards
    1

    What is the last digit of 2^1000

    I just want to know a good way to attack problems of this sort. I think I have the idea somewhat, however some expertise on the matter could go a long way in hammering home these ideas.


    What is the last digit of 2^1000?

    I have it as being 6 because 2^1000 is also 2^(2^3)(5^3)which is simply 2^8(125) which is 256^125. Any 6 when put to any power will always have a six as the last digit. That is how I got my answer but I am sure that there is a much more formal and proper way to write this out. Thanks a lot in advance for any help on this.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,663
    Thanks
    603
    Hello, padsinseven!

    I think your solution is excellent.
    It can be streamlined a bit . . .


    What is the last digit of 2^{1000}?

    2^{1000} \:=\:(2^4)^{250} \:=\:(16)^{250}

    Fact: any number ending in 6, when raised to a power, will end in 6.

    Therefore, the last digit is 6.

    Follow Math Help Forum on Facebook and Google+

  3. #3
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1
    Another sobervation you could make is the last digit of the 2 powers.

    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

    See, the last digit has the pattern: 2,4,8,6. Then it starts over.

    Since 4 divides into 1000 evenly, the last digit is 6.

    Suppose it were 2^2157112678. What's the last digit?.

    4 divides 2157112678 with 2 as a remainder. Therefore, the last digit is 4.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Digit sum & digit product of number x
    Posted in the Algebra Forum
    Replies: 1
    Last Post: January 19th 2011, 08:07 AM
  2. [SOLVED] 1!+2!+...+1000! \equiv x (mod 10)
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: June 26th 2010, 07:42 PM
  3. 1-1000 how many 3's?
    Posted in the Math Topics Forum
    Replies: 1
    Last Post: February 5th 2009, 05:58 PM
  4. decimal digit as final digit
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: February 25th 2008, 07:18 PM
  5. 1 coin tossed 1000 times vs 1000 coins
    Posted in the Advanced Statistics Forum
    Replies: 14
    Last Post: December 20th 2007, 06:55 PM

Search Tags


/mathhelpforum @mathhelpforum