Question

Given any integers a,b and c, if a-b is odd and b-c is odd, what

can you say about the parity of a-c? Write a formal proof and use
the definitions of odd or even.

Answer 1

When a-b and b-c are odd integers, a-c will always be even.

Step-by-step explanation:

To determine the parity of a-c, we can use the given information about a-b and b-c being odd. Let's assume a-b and b-c are odd integers.

1. Since a-b is odd, we can write it as a-b = 2k+1, where k is an integer.
2. Similarly, b-c can be expressed as b-c = 2m+1, where m is an integer.

Now, we can find a-c by combining the two equations as:

a-c = (a-b) + (b-c) = (2k+1) + (2m+1) = 2(k+m+1).

Here, k+m+1 is an integer, as the sum of two integers is always an integer. Therefore, a-c is even.