Question

Find the smallest distinct positive numbers that provide a counterexample to show the statement is false,The sum of any two different odd numbers is divisible by 4.The counterexample is

Answer 1

ANSWER

1 and 5

EXPLANATION

We want to find the two smallest distinct positive numbers that provide a counterexample for:

This means that we want to find the smallest two distinct odd numbers that make the statement false.

The statement means that:

$\frac{x+y}{4}=n$

where x and y are odd numbers and n is an integer.

The first two smallest distinct odd numbers are 1 and 3, so we have:

$\begin{gathered} \frac{1+3}{4}=\frac{4}{4} \\ =1 \end{gathered}$

As we can see, the sum is divisible by 4.

Let us move to the next two, 1 and 5:

$\frac{1+5}{4}=\frac{6}{4}$

As we can see, 6 is not divisible by 4.

This means that the smallest distinct positive numbers that provide a counterexample to show that the statement is false are 1 and 5.