Which of the following pairs of numbers are relatively prime? Show the calculations that led to your conclusions

$$\begin{gathered} \mathbf{a}.\mathbf{1274}\mathbf{and}\mathbf{10505} \\ \mathbf{b}\mathbf{.}\mathbf{7289}\mathbf{and}\mathbf{8029} \end{gathered}$$

Relative primeis a pair of integers is said to be relative prime if their common factor is $1$.

(a)The pair of numbers $1274and10505$are relatively prime.

Explanation:

$$\begin{gathered} Applyfirstno 1274=2x7x7x13x1. \\ Another{2}^{nd}no 10505=5x11x191x1. \end{gathered}$$

Another common element between both the two groups has now been identified.$1274and10505is1$$1274and10505is1$.

Here is,$GCD\left(1274,10505\right)=1$.

Each GCD would have to have relative prime.

Therefore, the pairs of numbers $1274and10505$are relative prime.

(b)The pair of numbers $7289and8029$are not relatively prime.

Explanation:

$$\begin{gathered} UseFirst7289=37x197x1. \\ AnotherNumber8029=7x31x37x1. \end{gathered}$$

The similarities between the two groups have now been identified.$7289and8029are37and1$.

Thus $GCD\left(7289,8029\right)=37$

The GCD must be$1$according to the concept of relative prime.

Hence, the numerals $7289through8029$are not very prime