Question

The sum of two numbers is 324 and their HCF is 9. Find the numbers.

Answer 1

Hint: Here, we will assume that the two numbers are \[x\] and \[y\]. Then we will add them to form the first equation and as we are given that 9 is their HCF, so each number must be divisible by 9, where \[x\] and \[y\] are coprime. Then we will take \[x = 9a\] and \[y = 9b\] in the obtained equation to find the numbers that are just co-prime with each other to find the required numbers.

Complete step by step answer:

We are given that the sum of two numbers is 324 and their HCF is 9.
Let us assume that the two numbers are \[x\] and \[y\].
Since we know that the sum of two numbers is 324, we have
\[ \Rightarrow x + y = 324{\text{ .......eq.(1)}}\]
We are given that 9 is their HCF, so each number must be divisible by 9, where \[x\] and \[y\] are coprime.
So taking \[x = 9a\] and \[y = 9b\] in the equation (1), we get
\[ \Rightarrow 9a + 9b = 324\]
Dividing the above equation by 9 on both sides, we get
\[
   \Rightarrow \dfrac{{9a}}{9} + \dfrac{{9b}}{9} = \dfrac{{324}}{9} \\
   \Rightarrow a + b = 36 \\
 \]
So, there are two pairs \[\left( {x,y} \right)\], which satisfies that both the numbers are co-prime with each other is \[\left( {13,23} \right)\] and \[\left( {5,31} \right)\].
Multiplying the above coordinate by 9 to find the final answer, we get
\[
   \Rightarrow \left( {9 \times 13,9 \times 23} \right) \\
   \Rightarrow \left( {117,207} \right) \\
 \]
and
\[
   \Rightarrow \left( {9 \times 5,9 \times 31} \right) \\
   \Rightarrow \left( {45,279} \right) \\
 \]
Thus, the possible answers are \[\left( {117,207} \right)\] and \[\left( {45,279} \right)\].

Note: In this question, we have to know that if the HCF of numbers is that each number must be divisible by it, where \[x\] and \[y\] are coprime. Also, we are supposed to write the values properly to avoid any miscalculation. You can check the results by adding them which will give 324.