Question

Find the HCF $\times $ LCM for the numbers 100 and 190.

Answer 1

Hint: To find HCF $\times $ LCM for the numbers 100 and 190, we have to use the property of LCM and HCF which states that the product of HCF and LCM of two numbers is equal to the product of these two numbers, provided the numbers are natural numbers.

Complete step by step answer:
We have to find HCF $\times $ LCM for the numbers 100 and 190. Let us recollect what LCM and HCF are. The Least Common Multiple (LCM) of numbers, say a and b, is the smallest positive integer that is evenly divisible by both a and b. The Highest Common Factor(HCF) is the greatest number which divides each of the two or more numbers.
We know that there are a lot of properties related to LCM and HCF. One of these is that the product of HCF and LCM of two numbers is equal to the product of these two numbers, provided the numbers are natural numbers.
$\begin{align}
  & \Rightarrow \text{HCF}\left( 100,90 \right)\times \text{LCM}\left( 100,90 \right)=100\times 190 \\
 & \Rightarrow \text{HCF}\left( 100,90 \right)\times \text{LCM}\left( 100,90 \right)=19000 \\
\end{align}$

Hence, HCF $\times $ LCM for the numbers 100 and 190 is 19000.

Note: Students must be thorough with the properties of LCM and HCF. If they do not remember the formula $\text{HCF}\left( a,b \right)\times \text{LCM}\left( a,b \right)=a\times b$ , they can follow the alternate method shown below to find HCF $\times $ LCM for the numbers 100 and 190.
We have to first find the HCF of 100 and 190. Let us use the ladder method(division method).
$\begin{align}
  & 2\left| \!{\underline {\,
  100,190 \,}} \right. \\
 & 5\left| \!{\underline {\,
  50,95 \,}} \right. \\
 & \text{ }\text{ }\text{ }10,19 \\
\end{align}$
Therefore, we can write the HCF as $2\times 5=10$ .
Now, let us find the LCM of 100 and 190.
$\begin{align}
  & \text{ }\text{ } 2\left| \!{\underline {\,
  100,190 \,}} \right. \\
 & \text{ }\text{ }5\left| \!{\underline {\,
  50,95 \,}} \right. \\
 & \text{ }\text{ }2\left| \!{\underline {\,
  10,19 \,}} \right. \\
 & \text{ }\text{ }5\left| \!{\underline {\,
  5,19 \,}} \right. \\
 & 19\left| \!{\underline {\,
  1,19 \,}} \right. \\
 & \text{ }\text{ }\text{ }\text{ }\text{ }1,1 \\
\end{align}$
Therefore, we can write the LCM of 100 and 190 as $2\times 5\times 2\times 5\times 19=1900$ .
Now, we have to multiply HCF and LCM.
$\Rightarrow \text{HCF}\times \text{LCM}=1900\times 10=19000$