Question

If the HCF of \[35 \] and \[45 \] is \[5 \] , LCM of \[35 \] and \[45 \] is \[63 \times a \] , then find the value of \[a \] .

Answer 1

Hint: The given question involves the arithmetic operations like addition/ subtraction/ multiplication/ division. We need to know the basic formula with the involvement of HCF and LCM to make an easy calculation. Also, we need to know how to separate a variable from constants to solve these types of questions.

Complete step by step solution:
In this question we have,
HCF of \[35 \] and \[45 \] is \[5 \]
And
LCM of \[35 \] and \[45 \] is \[63 \times a \] .
So, it can also be written as,
 \[HCF\left( {35,45} \right) = 5 \to \left( 1 \right) \]
 \[LCM\left( {35,45} \right) = 63 \times a \to \left( 2 \right) \]
We know that
If \[m,n \] are two positive numbers then,
 \[HCF\left( {m,n} \right) \times LCM\left( {m,n} \right) = m \times n \to \left( 3 \right) \]
By comparing the equation \[\left( 1 \right)\& \left( 2 \right) \] with the equation \[\left( 3 \right) \] , we get
 \[
  HCF\left( {m,n} \right) = HCF\left( {35,45} \right) \\
  LCM\left( {m,n} \right) = LCM\left( {35,45} \right) \\
  \]
So, we get
 \[
  m = 35 \\
  n = 45 \\
  \]
So, the equation \[\left( 3 \right) \] becomes,
 \[\left( 3 \right) \to HCF\left( {m,n} \right) \times LCM\left( {m,n} \right) = m \times n \]
 \[HCF\left( {35,45} \right) \times LCM\left( {35,45} \right) = 35 \times 45 \]
So, we get
 \[HCF\left( {35,45} \right) \times LCM\left( {35,45} \right) = 1575 \to \left( 4 \right) \]
Let’s substitute the equation \[\left( 1 \right)\& \left( 2 \right) \] in the equation \[\left( 4 \right) \] , we get
 \[\left( 4 \right) \to HCF\left( {35,45} \right) \times LCM\left( {35,45} \right) = 1575 \]
 \[5 \times 63 \times a = 1575 \]
Let’s separate the term \[a \] from the above equation we get,
 \[a = \dfrac{{1575}}{{5 \times 63}} = \dfrac{{1575}}{{315}} \]
By using calculator we get,
 \[a = \dfrac{{1575}}{{315}} = 5 \]
So, the final answer is,
 \[a = 5 \]

Note: This question describes the arithmetic operations like addition/ subtraction/ multiplication/ division. We would create an equation from the given data to solve these types of questions. Also, we would separate the unknown term from the known terms to solve the given problem. Also, note that when we move one term from the left-hand side to the right-hand side of the equation the arithmetic functions can be modified as follows,
The addition operation can be converted into the subtraction process.
The subtraction process can be converted into the addition process.
The multiplication process can be converted into the division process.
The division process can be converted into a multiplication process