Question

If \[\left( 24,92 \right)=24m+92n\], then, \[\left( m,n \right)\]is
\[\begin{align}
  & \left( a \right)\left( -4,3 \right) \\
 & \left( b \right)\left( -1,4 \right) \\
 & \left( c \right)\left( 4,-1 \right) \\
 & \left( d \right)\left( 4,-3 \right) \\
\end{align}\]

Answer 1

Hint: Here we have to find the HCF of \[24\] and \[92\], because \[\left( 24,92 \right)\] means the HCF of \[24\] and \[92\]. For this, we have to write the factors of \[24\] and \[92\] first, and then find the common factors and then find the value of \[\left( m,n \right)\].

Formulae Used:
 In order to find the HCF, list the factors of the numbers \[24\] and \[92\], and the greatest common factors form the HCF.
Where the HCF is equal to the condition given in the question
\[\left( 24,92 \right)=24m+92n\]
Put all the options in the equation to find the correct values of \[m\] and \[n\].

Complete step by step solution:
First of all, as we can see that \[\left( 24,92 \right)\] means the HCF of the two numbers \[24\] and \[92\] so, let us find the HCF of \[24\]and\[92\], by writing the factors,
The factors of \[24\]are: \[2\times 2\times 2\times 3\]
And the factors of\[92\]are: \[2\times 2\times 23\]
Therefore, the common factors are \[2\times 2\]
So, the HCF is \[4\].
As given in the question, the HCF equals \[24m+92n\]
\[\begin{align}
  & \therefore 24m+92n=4 \\
 & \Rightarrow 4\left( 6m+23n \right)=4 \\
 & \Rightarrow 6m+23n=1 \\
\end{align}\]
To satisfy the equation, we put all the four options one by one and check for the correct one.
So, For option \[\left( a \right)\left( -4,3 \right)\]
\[\begin{align}
  & \Rightarrow 6\left( -4 \right)+23\left( 3 \right) \\
 & \Rightarrow -24+69 \\
 & \Rightarrow 45 \\
\end{align}\]
For option \[\left( b \right)\left( -1,4 \right)\]
\[\begin{align}
  & \Rightarrow 6\left( -1 \right)+23\left( 4 \right) \\
 & \Rightarrow -6+92 \\
 & \Rightarrow 86 \\
\end{align}\]
For option \[\left( c \right)\left( 4,-1 \right)\]
\[\begin{align}
  & \Rightarrow 6\left( 4 \right)+23\left( -1 \right) \\
 & \Rightarrow 24-23 \\
 & \Rightarrow 1 \\
\end{align}\]
And for option \[\left( d \right)\left( 4,-3 \right)\]
\[\begin{align}
  & \Rightarrow 6\left( 4 \right)+23\left( -3 \right) \\
 & \Rightarrow 24-69 \\
 & \Rightarrow 45 \\
\end{align}\]

Hence, option \[\left( c \right)\left( 4,-1 \right)\] satisfies the equation, Hence it is correct.
\[\Rightarrow \left( m,n \right)=\left( 4,-1 \right)\].

Note: To solve such a problem, we find the HCF of the two numbers, and substitute the values of \[m\] and\[n\] given in the options and the calculated HCF , into the equation and find the option that fits the equation and that option is taken as the correct option .