relatively prime

[logistic_guy]

Find all positive integers less than \(\displaystyle 10\) that are relatively prime to it.

 

[logistic_guy]

Find all positive integers less than \(\displaystyle 10\) that are relatively prime to it.

The numbers \(\displaystyle \textcolor{blue}{1}\), \(\displaystyle \textcolor{blue}{3}\), \(\displaystyle \textcolor{blue}{7}\), and \(\displaystyle \textcolor{blue}{9}\) are all relatively prime to the number \(\displaystyle \textcolor{red}{10}\).

But why?

The idea is simple when the \(\displaystyle \text{GCD}\) of two integers is \(\displaystyle 1\), then they are relatively prime.

For example,
\(\displaystyle 10 = 1 \times 2 \times 5\)
\(\displaystyle 9 = 1 \times 3 \times 3\)
Then
\(\displaystyle \text{GCD(10,9)} = 1\)

But
\(\displaystyle 10 = 1 \times 2 \times 5\)
\(\displaystyle 5 = 1 \times 5\)
Then
\(\displaystyle \text{GCD(10,5)} = 5 \neq 1\)

This is one example of why \(\displaystyle 9\) is relatively prime to \(\displaystyle 10\) but \(\displaystyle 5\) not!

 

[pka]

Find all positive integers less than \(\displaystyle 10\) that are relatively prime to it.

Note that [imath]10[/imath] is an even integer therefore cannot be relatively prime with any other positive even integer.

 

[logistic_guy]

Note that [imath]10[/imath] is an even integer therefore cannot be relatively prime with any other positive even integer.

Yeah I have noticed this fact as all positive even integers share at least the number \(\displaystyle 2\).