Question

On a morning walk, three persons step off together and their steps measure 40cm, 42cm and 45cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?

Answer 1

Hint: LCM determines the least multiple of the given numbers. Take LCM of the measure of one step of one person, use the factorization method to get the factors. Multiply them to get the LCM.

Complete step-by-step answer:
As we know LCM stands for least common multiple i.e. LCM of the given numbers will be the least number which is divisible by given numbers. Now coming to the question, it is given that the length of the steps by three persons are 40cm, 42cm and 45cm. So, they will meet at a certain distance which should be a multiple of given numbers because the length covered by each person will be a multiple of measure of one step of each person.
Hence, the distance at which all the three persons will meet for the very first time is the LCM of the measures of one step of each person i.e. LCM of 40, 42, 45. So, we can find the LCM of 40, 42, 45 by doing prime factorization of them.
$\begin{align}
  & 2\left| \!{\underline {\,
  40,42,45 \,}} \right. \\
 & 2\left| \!{\underline {\,
  20,21,45 \,}} \right. \\
 & 2\left| \!{\underline {\,
  10,21,45 \,}} \right. \\
\end{align}$
$\begin{align}
  & 3\left| \!{\underline {\,
  5,21,45 \,}} \right. \\
 & 3\left| \!{\underline {\,
  5,7,15 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5,7,5 \,}} \right. \\
 & 7\left| \!{\underline {\,
  1,7,1 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1,7,1 \,}} \right. \\
\end{align}$
So, LCM of 40, 42, 45 can be given as
$\begin{align}
  & LCM=2\times 2\times 2\times 3\times 3\times 5\times 7 \\
 & LCM=2520 \\
\end{align}$
Hence, LCM of 40, 45, 45 is 2520.
So, all three persons will meet at 2520m distance for the first time or in other words the minimum distance each should walk so that each can cover the same distance in complete steps is 2520m.

Note: One may go wrong with this type of questions, as one may calculate HCF of the lengths in place of LCM of them. It is common sense that HCF will be lower than the given numbers. So, it cannot represent the required distance. LCM can also be calculated by individual prime factorization of the numbers as well.
$\begin{align}
  & 40=2\times 2\times 2\times 5 \\
 & 42=2\times 3\times 7 \\
 & 45=3\times 3\times 5 \\
\end{align}$
LCM = multiply the common numbers at once and hence with other numbers.
$\begin{align}
  & LCM=2\times 3\times 3\times 5\times 2\times 2\times 7 \\
 & =2520 \\
\end{align}$