Question

Which positive integers less than 12 are relatively prime to 12?

Short answer

1,5,7,11

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 12 and every integer less than 12.

$$\begin{gathered} \gcd (12,1)=1 \\ \gcd (12,2)=2 \\ \gcd (12,3)=3 \\ \gcd (12,4)=4 \\ \gcd (12,5)=5 \\ \gcd (12,6)=6 \\ \gcd (12,7)=7 \\ \gcd (12,8)=8 \\ \gcd (12,9)=9 \\ \gcd (12,10)=10 \\ \gcd (12,11)=11 \end{gathered}$$

The positive integers less than 12 that are relatively prime to 12 are then all integers $a$for which $\gcd =\left(12,\mathrm{a}\right)=1$

$a=1,5,7,11$