Question

If the HCF of $85$ and $153$ is expressible in the form $85n-153,$ then value of n is :
$
  A.\,{\text{ }}3 \\
  B.{\text{ 2}} \\
  {\text{C}}{\text{. 4}} \\
  {\text{D}}{\text{. 1}} \\
 $

Answer 1

Hint: Highest common factor (HCF) is the greatest number which divides each of the two or more numbers. It is also known as the Greatest Common Measure (GCM) and the Greatest Common Divisor (GCD). Here we will find the HCF of both the given numbers and will compare it with the given form to find the unknown term.

Complete step-by-step answer:
Now, the Highest common factor (HCF) by the factorization method –
$\Rightarrow 85=17\times 5\ \text{ }......\text{(a)}$
Similarly for the other given number –
$\begin{align}
  & 153=17\times 9 \\
 & 153=17\times 3\times 3\text{ }.........\text{(b)} \\
\end{align}$
From the equations (a) and (b) –
The common factors of both the given numbers $85\,\text{and 153 is 17}$
Hence, the HCF of the given two numbers $85\,\text{and 153 is 17}$ …………………. (c)
Now, put the value of the equation (c) in the given expression
$\Rightarrow 85n-153=17$
Simplify using the basic mathematical operations, by the property when the term is moved from the right hand side to left hand side then the negative term is changed to positive and vice-versa.
$\begin{align}
  & 85n=17+153 \\
 & 85n=170 \\
\end{align}$
U se the identity, when the term in multiplicative at one side is moved to the other side, it goes to the division and vice-versa.
$\begin{align}
  & n=\dfrac{170}{85} \\
 & n=2 \\
\end{align}$
Therefore, the required solution - If the HCF of $85$ and $153$ is expressible in the form $85n-153,$ then value of n is $2$

So, the correct answer is “Option B”.

Note: We can find the HCF of any given number by two methods.
\[1)\] Prime factorization method and $2)$ the division method. Also, go through the difference between LCM (Least common Multiple) and HCF (Highest common multiple) and methods to find to solve these types of problems.