number problem

prove that every set of seven distinct integers contarins a pair whose sum or difference is a multiple of 10.

i know that the last digit of any number has to end with 0,1,2,3,4,5,6,7,8,9. is this how i should go above proving this? i dont understand how the difference can be a multiple of 10. please help.

thank you

Quote:

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Originally Posted by demon1

prove that every set of seven distinct integers contarins a pair whose sum or difference is a multiple of 10.

i know that the last digit of any number has to end with 0,1,2,3,4,5,6,7,8,9. is this how i should go above proving this? i dont understand how the difference can be a multiple of 10. please help.

thank you

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Just start by listing the possibilities. If two numbers have the same last digit their difference is divisible by 10 so we may assume all the numbers end in different digits. So the digits could end as:
..0
..1
..2
..3
..4
..5
..6

But 4 + 6 = 10 so the numbers ending in 4 and 6 add to a number divisible by 10. So that's out. The next possibility is:
..0
..1
..2
..3
..4
..5
..7

But the numbers ending in 3 and 7 add to a number divisible by 10, etc.

See if you can generalize this argument and see if you can explain why if you have a list of 6 numbers this theorem isn't true.

-Dan