Question

Determine whether the integers in each of these sets are pair wise relatively prime.
$$\begin{array}{ll}\text{a)}11,15,19& \text{b)}14,15,21\\ \text{c)}12,17,31,37& \text{d)}7,8,9,11\end{array}$$

Answer 1

Checking if the numbers are relatively prime:

$2$ numbers are relatively prime if they do not have any common factor other than $\pm 1$

$a)$ $11,15,19$

Writing their prime factorization:

$$\begin{gathered} 11=11 \\ 15=3\times 5 \\ 19=19 \end{gathered}$$

Any pair of these numbers do not have any common factor, hence they are relatively prime

$b)$$14,15,21$

Writing their prime factorization:

$$\begin{gathered} 14=7\times 2 \\ 15=3\times 5 \\ 21=3\times 7 \end{gathered}$$

$15,21$ has a common factor of $3$

$14,21$ has a common factor of $7$

Hence they are not relatively prime

$c)$$12,17,31,37$

Writing their prime factorization:

$$\begin{gathered} 12=3\times 2^2 \\ 17=17 \\ 31=31 \\ 37=37 \end{gathered}$$

Any pair of these numbers do not have any common factor, hence are relatively prime

$d)$$7,8,9,11$

Writing their prime factorization:

$$\begin{gathered} 7=7 \\ 8=2^3 \\ 9=3^2 \\ 11=11 \end{gathered}$$

Any pair of these numbers do not have any common factor, hence are relatively prime.