Question

How many times is the HCF of 24, 64, 32, 72 is contained in the LCM?
(a) 36
(b) 60
(c) 72
(d) 42

Answer 1

Hint: In this type of question we have to use the concept of finding HCF and LCM of the given numbers. To solve the given question first we find all the factors of the given numbers. After that we will find the HCF and LCM of the given numbers and then we divide LCM by HCF to find the required result.

Complete step by step answer:
Now, we have to find the number of times which the HCF of 24, 64, 32, 72 is contained in the LCM.
Let us find the HCF of all the given numbers by using common division method as follows:
\[\begin{align}
  & \Rightarrow 2\left| \!{\underline {\,
  24,64,32,72 \,}} \right. \\
 & \text{ }2\left| \!{\underline {\,
  12,32,16,36 \,}} \right. \\
 & \text{ }2\left| \!{\underline {\,
  6,16,8,18 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  3,8,4,9 \,}} \right. \\
\end{align}\]
Division of 24, 64, 32 and 72 by third 2 will leave 3, 8, 4 and 9 as their remainders respectively. 3, 8, 4 and 9 do not have a common prime factor.
Hence, the HCF of 24, 64, 32 and 72 is \[2\times 2\times 2=8\].
Now, we will find the LCM of all the numbers by using common division method as follows:
\[\begin{align}
  & \Rightarrow 2\left| \!{\underline {\,
  24,64,32,72 \,}} \right. \\
 & \text{ }2\left| \!{\underline {\,
  12,32,16,36 \,}} \right. \\
 & \text{ }2\left| \!{\underline {\,
  6,16,8,18 \,}} \right. \\
 & \text{ 2}\left| \!{\underline {\,
  3,8,4,9 \,}} \right. \\
 & \text{ }2\left| \!{\underline {\,
  3,4,2,9 \,}} \right. \\
 & \text{ }2\left| \!{\underline {\,
  3,2,1,9 \,}} \right. \\
 & \text{ }3\left| \!{\underline {\,
  3,1,1,9 \,}} \right. \\
 & \text{ }3\left| \!{\underline {\,
  1,1,1,3 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1,1,1,1 \,}} \right. \\
\end{align}\]
Hence, the LCM of 24, 64, 32 and 72 is \[2\times 2\times 2\times 2\times 2\times 2\times 3\times 3=576\]
Hence, the HCF and LCM of 24, 64, 32 and 72 are 8 and 576 respectively.
Thus, \[\dfrac{576}{8}=72\] times 8 is contained in the 576.
Hence, 72 times HCF is contained in the LCM.

So, the correct answer is “Option c”.

Note: One of the students may solve this question as follows:
\[\begin{align}
  & \Rightarrow \text{Factors of 24 = }1,2,3,4,6,8,12,24 \\
 & \Rightarrow \text{Factors of 64 = }1,2,4,8,16,32,64 \\
 & \Rightarrow \text{Factors of 32 = }1,2,4,8,16,32 \\
 & \Rightarrow \text{Factors of 72 = }1,2,3,4,6,8,9,12,18,24,32,72 \\
 & \Rightarrow \text{Common Factors = 1,2,4,8} \\
 & \Rightarrow \text{HCF = 8} \\
\end{align}\]
We know that the product of the numbers is equal to the product of HCF and LCM of the given numbers. But as here, 32 and 64, 24 and 72 are multiples of each other so that we consider multiplication of 64 and 72 only. Thus we have,
\[\begin{align}
  & \Rightarrow \text{Product of 72 and 64 = HCF }\times \text{ LCM} \\
 & \Rightarrow \text{72}\times \text{64 = 8 }\times \text{ LCM} \\
 & \Rightarrow \dfrac{4608}{8}=\text{ LCM} \\
 & \Rightarrow \text{576 = LCM} \\
\end{align}\]
 Hence, the HCF and LCM of 24, 64, 32 and 72 are 8 and 576 respectively.
Thus, \[\dfrac{576}{8}=72\] times 8 is contained in the 576.
Hence, 72 times HCF is contained in the LCM.
Thus, option (c) is the correct option.