Question

If HCF of 144 and 180 is expressed in the form 13m – 16. Find the value of m.

Answer 1

Hint: We have been given two numbers and we have to express the HCF in a particular form. So, first of all we have to find the HCF of 144 and 180 and we can use the prime factorization method. Then we can equate the HCF to 13m – 16. Then, by substituting the value of HCF and simplifying it, we can find the value of m.

Complete step-by-step answer:
Let us first look into what prime factorization is and it is used to break down or express a given number as a product of prime numbers.
Moreover, if a prime number occurs more than once in the factorization, it is usually expressed as a whole number greater than 1 that is only divisible by 1 and itself and another way to say is that a prime number has exactly two factors, namely 1 and itself.
Then, we have been told that the HCF of 144 and 180 is expressed in form 13m – 16. We need to find the value of m. Let us use the prime factorization method to find the HCF of 144 and 180.\[\begin{align}
  & 180=2\times 2\times 3\times 3\times 5={{2}^{2}}\times {{3}^{2}}\times 5 \\
 & 144=2\times 2\times 2\times 2\times 3\times 3={{2}^{2}}\times {{2}^{2}}\times {{3}^{2}} \\
\end{align}\]
From both 144 and 180, \[\left( {{2}^{2}}\times {{3}^{2}} \right)\] is common.
So, we get the HCF (144, 180) = \[{{2}^{2}}\times {{3}^{2}}=4\times 9=36\]
Hence, we got HCF = 36
Now it is told to us that HCF is in terms of 13m – 16.
Now, we are given in the question as HCF = 13m – 16 and then by equating is to 36, we get:
36 = 13m – 16
Now let us simplify it and get the value of m.
13m = 36 + 16
\[\begin{align}
  & \Rightarrow m=\dfrac{52}{13} \\
 & \Rightarrow m=4 \\
\end{align}\]
Hence, we got the value of m = 4.

Note: We can find the LCM of the same numbers 180 and 144 by prime factorization method.$\begin{align}
  & 180=2\times 2\times 3\times 3\times 5={{2}^{2}}\times {{3}^{2}}\times 5 \\
 & 144=2\times 2\times 2\times 2\times 3\times 3={{2}^{4}}\times {{3}^{2}} \\
\end{align}$
From the above LCM is the common terms multiplied with the rest of the terms.
LCM=\[\left( {{2}^{2}}\times {{3}^{2}} \right)\times 4\times 5=720\]
LCM =720