Three points A, B and C with natural coordinates were marked on the coordinate ray.

| January 11, 2021 | education

Three points A, B and C with natural coordinates were marked on the coordinate ray. Prove that at least one of the mid-segments with endpoints at these points has a natural coordinate.

Let the coordinates of points A, B and C be natural numbers a, b, c.
Find the coordinates of the midpoints of the segments with the ends at points A, B and C.
For segment AB: x1 = (b – a) / 2.
For segment BC: x2 = (c – a) / 2.
For the segment AC: x3 = (c – a) / 2.
Since a, b and c are natural numbers, and when divided by 2, a natural number can either be divisible by 2, or give when divided by 2 in the remainder of 1, i.e. be either an even number or an odd number, then at least 2 of the three numbers a, b, c will be either even or odd.
Therefore, the difference between these two numbers will be an even number.
This means that at least one of the values ​​x1, x2, x3 is a natural number, which is what we had to prove.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.