Question

let gcd (a,4)=2 and gcd(a,b)=1. If 4>a>b and a, b are natural numbers, then the value of a and b are ?​

Answer 1

Given:

$gcd (a,4)=2$ and $gcd (a, b)=1$ where, 4>a>b and a, b ∈ Ν.

To Find:

Value of  and b are ?

Solution:

Since, $gcd (a,4)=2$  and  $gcd (a, b)=1$ where, 4>a>b and a, b ∈ Ν

Therefore, for $gcd(a, 4)=2$ .

From division algorithm;

⇒ 2 divides a and 4 then there exist m, n such that-

$\[\begin{gathered} a = m \times 2 \hfill \\ a = 2m \hfill \\ \end{gathered} \]$      and        $\[\begin{gathered} 4 = n \times 2 \hfill \\ n = 2 \hfill \\ \end{gathered} \]$    ....(1)

also, $gcd (a, b)=1$

⇒1 divides a and b then there exist m, n such that-

$\[\begin{gathered} a = m \times 1 \hfill \\ a = m \hfill \\ \end{gathered} \]$        and       $\[\begin{gathered} b = n \times 1 \hfill \\ b = 2 \times 1 = 2 \hfill \\ \end{gathered} \]$      ....(2)

From equation 1st and 2nd , we get

m=3 , n = 2

Thus, a =3 or 6 but it is given that -  4>a>b

⇒  a = 3 .        (4>a=3>2 )

Hence, a=3 and b=2.