1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Sum divisable by 5.

  1. Nov 7, 2004 #1


    User Avatar
    Science Advisor
    Homework Helper

    The question asks me to prove that if I have 5 arbitrary natural numbers to show that one of the sums (in order) is divisible by 5. So say the numbers are [itex]a_1,a_2,a_3,a_4,a_5[/itex] some examples would be:

    [tex]a_1 + a_2 + a_3[/tex]


    [tex]a_3 + a_4[/tex]


    My first thought was to consider the remainder upon division 5 and then start identifying all the possible combinations where the sum is divisible by 5 and show there are no more left. However when starting this I realised this was actually a lot of work and there must be some simpler way. Can anyone else point me in a different direction?
  2. jcsd
  3. Nov 17, 2004 #2


    User Avatar
    Homework Helper

    I don't know what in order means so I hope it does not matter.
    consider the sums
    consider their remainders upon division by 5
    There are 5 possible remainders {0,1,2,3,4} and 5 sums so either a sum is divisible by 5 (remainder=0) or of two of the sums remainders are equal.
    if 2 sums remainders are equal the difference of the sums is itself a sum and is divisible by 5
    so for example if (a1,a2,a3,a4,a5,n,m,r are natural numbers)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook