Problem

Dear Friends and Visitors of the site!

Try our February 2018 Problem:

Given a set of 11 natural numbers, such that the sum of any ten of them is a multiple of 7.

Show that all these numbers are multiples of 7.

Waiting for your solutions.

Serge Hazanov

 

Answer Submission Is Not Available

Form is valid through February 2018

Solution

Dear visitors of the site,

many thanks for your effort and solutions.

The first to deliver correct solutions were Harry and James.

Congratulations. Well done!

And here is my solution:

Take 2 numbers and two sets of 10 :

One without the first number and the second without the second number.

Easy to see that both numbers are equal by mod 7.

Thus all the numbers are equal by mod 7.

As the sum of any 10 must me a multiple of 7, any number from the set is a multiple of 7.

Thank you again and see you back in April.

Yours,

Serge Hazanov