Question

Which positive integers less than 30 are relatively prime to 30?

Short answer

1, 7, 11, 13, 17, 19, 23, 29

Step-by-step solution

Step 1

The integers $a$ and $b$ are relatively prime, if their greatest common divisor is 1.

Let us determine the greatest common divisor of 30 and every integer less than 30.

$\begin{array}{c}\gcd (30,1)=1\\ \gcd (30,2)=2\\ \gcd (30,3)=3\\ \gcd (30,4)=4\\ \gcd (30,5)=5\\ \gcd (30,6)=6\\ \gcd (30,7)=7\\ \gcd (30,8)=8\\ \gcd (30,9)=9\\ \gcd (30,10)=10\\ \gcd (30,11)=1\\ \gcd (30,12)=6\\ \gcd (30,13)=1\\ \gcd (30,14)=2\\ \gcd (30,15)=5\\ \gcd (30,16)=2\\ \gcd (30,17)=1\\ \gcd (30,18)=6\\ \gcd (30,19)=1\\ \gcd (30,20)=10\end{array}$

Step 2

$\begin{array}{r}\gcd (30,21)=3\\ \gcd (30,22)=2\\ \gcd (30,23)=1\\ \gcd (30,24)=6\\ \gcd (30,25)=5\\ \gcd (30,26)=2\\ \gcd (30,27)=3\\ \gcd (30,28)=2\\ \gcd (30,29)=1\end{array}$

The positive integers less than 30 that are relatively prime to 30 are then all integers $a$for which $\gcd =\left(30,1\right)=1$.

$a=1,7,11,13,17,19,23,29$