Question

Find the HCF of 96 and 404 by prime factorization method. Hence, find their LCM.

Answer 1

Hint: In the prime factorization method, we write the numbers as a product of their prime factors. Then the product of factors that are common to both numbers is the highest common factor.

Complete step by step answer:
In this problem we will prime factorize 96 and 404. The prime factorization of these two number has been done below:

$\begin{align}

  & 96=2\times 2\times 2\times 2\times 2\times 3 \\

 & 404=2\times 2\times 101

\end{align}$

Here we can see that the $2\times 2=4$ is the highest common factor of 96 and 404.
To find the LCM (Lowest Common Multiple) we need not calculate it from the beginning but by a simple rule given below:

If a and b are two numbers then

$a\times b=(H.C.F\ of\ a\ and\ b)\times (L.C.M\ of\ a\ and\ b)$

Therefore, after rearrangement we can write

Therefore, L.C.M of 96 and 404 is given by

$\begin{align}

  & L.C.M\ of\ 96\ and\ 404\ =\ \dfrac{96\times 404}{H.C.F\ of\ 96\ and\ 404} \\

 & =\dfrac{96\times 404}{4} \\

 & =96\times 101 \\

 & =9696

\end{align}$

Hence, L.C.M of 96 and 404 is 9696.



Note: We can verify the result obtained by evaluating the L.C.M 96 and 404 by prime factorization method

 $\begin{align}

  & 96=2\times 2\times 2\times 2\times 2\times 3 \\

 & 404=2\times 2\times 101

\end{align}$

Thus, L.C.M of 96 and 404 is $2\times 2\times 2\times 2\times 2\times 3\times 101=32\times 3\times 101=96\times 101=9696$

Hence, the result obtained from the method discussed in the solution agrees with the result obtained by evaluating the L.C.M from the prime factorization method. But, we can clearly see that the method we applied to find the L.C.M in the solution is much more efficient if we have large numbers having very few common factors. This way we can save ourselves from long multiplication.