Question

In a morning walk three persons step off together, their steps measure 80 cm, 85 cm and 90 cm respectively. What is the minimum distance each should walk so that they can cover the distance in complete steps?

Answer 1

Hint: For all three persons to cover the distance in complete steps, the distance should be a multiple of 80, 85 and 90. Find the LCM of 80, 85, 90.

Complete step-by-step answer:

It is given that the steps of the people measure 80 cm, 85 cm and 90 cm respectively. We need to find the minimum distance they can cover so that they have complete steps.
LCM or least common multiple is the smallest possible integer that is completely divisible by the given numbers.
To find the LCM of the given numbers, we need to factorize the three numbers together. Write the three numbers and first divide the numbers by the first prime number 2, keep dividing until all three numbers can’t be divided completely by 2. Then move on to the second prime number and repeat the same. Repeat the same with prime numbers until all three numbers are reduced to 1. Multiply all the factors to find the LCM.
\[\begin{gathered}
  {\text{ }}2\left| \!{\underline {\,
  {80,85,90} \,}} \right. \\
  {\text{ }}2\left| \!{\underline {\,
  {40,85,45} \,}} \right. \\
  {\text{ }}2\left| \!{\underline {\,
  {20,85,45} \,}} \right. \\
  {\text{ }}2\left| \!{\underline {\,
  {10,85,45} \,}} \right. \\
  {\text{ }}3\left| \!{\underline {\,
  {5,85,15{\text{ }}} \,}} \right. \\
  {\text{ }}3\left| \!{\underline {\,
  {5,85,5{\text{ }}} \,}} \right. \\
  {\text{ }}5\left| \!{\underline {\,
  {1,17,1{\text{ }}} \,}} \right. \\
  17\left| \!{\underline {\,
  {1,1,1{\text{ }}} \,}} \right. \\
\end{gathered} \]
LCM = \[2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 17\]
LCM = 12240
Hence, the minimum distance is 12240 cm.

Note: You can also directly attempt to find the minimum distance by considering the multiples of 80, 85 and 90, but solving by finding LCM is the best choice.