Question

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

Answer 1

Hint: In this question we are given measure of steps of three persons and we have to find the minimum distance each should walk so that he can cover the distance in complete step, that is, we need find the least common multiple of the measure of the steps. And we know the least common multiple (L.C.M) of any two non-zero integers $a$ and $b$ is the smallest positive integer that is divisible by both $a$ and $b$. So, we will find the L.C.M of the given measures of the steps.

Complete step-by-step answer:
In this question we are given that in a morning walk three persons step off together, their steps measure $80{\text{cm}}$, $85{\text{cm}}$ and $90{\text{cm}}$, respectively.
Now let $x,y,z$ be three persons and then
Measure of step of \[x\] is $80 {\text{cm}}$
Measure of step of \[y\] is $85 {\text{cm}}$
Measure of step of \[z\] is $90 {\text{cm}}$
Now according to the question, we have to find the minimum distance each should walk so that they should cover the distance in complete steps.
Now the question clearly suggests that we have to find the smallest common multiple of the measure of the steps.
Now let’s take a look on how to find the smallest common multiple-
The smallest common multiple is also known as least common multiple (L.C.M.) or the lowest common multiple.
Now the least common multiple of two non-zero integers $a$ and $b$ is the smallest positive integer that is divisible by both $a$ and $b$. Least common multiple of $a$ and $b$ is denoted by $LCM(a,b)$.
And similarly L.C.M of three non-zero integers is the smallest positive integer divisible by all the three integers.
Now the minimum distance each should walk so that they can cover the distance in complete steps is equal to-
$L.C.M{\text{ of measure of step of }}x{\text{, measure of step of }}y{\text{ and measure of step of }}z$
That is mathematically,
minimum distance each should walk so that they can cover the distance in complete steps $ = LCM\left( {80,85,90} \right)$ -(1)
now using the definition of L.C.M,
we can write that $LCM\left( {80,85,90} \right)$ is the smallest positive integer that is divisible by $80{\text{cm}}$, $85{\text{cm}}$ and $90{\text{cm}}$.
Now to find the smallest positive integer that is divisible by $80 {\text{cm}}$, $85 {\text{cm}}$ and $90 {\text{cm}}$, we have to find the factors of $80$, $85$ and $90$.
Now factors of-
$80 = 2 \times 2 \times 2 \times 2 \times 5$ -(2)
$85 = 17 \times 5$ -(3)
$90 = 2 \times 3 \times 3 \times 5$ -(4)
Now clearly from (2), (3) and (4), we can see that the smallest positive integer divisible by $80 {\text{cm}}$, $85 {\text{cm}}$ and $90 {\text{cm}}$ is $2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 17$.
Hence,
 $LCM\left( {80,85,90} \right) = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 17$
$LCM\left( {80,85,90} \right) = 12240$ -(5)
Now substituting value of $LCM\left( {80,85,90} \right)$ from (5) in (1), we get,
minimum distance each should walk so that they can cover the distance in complete steps $ = 12240 {\text{cm}}$

Note: In all the problems where we have to find the smallest integer divisible by all the given non-zero integers, we will always find the L.C.M of the given integers.
In the above question students generally make mistakes in the calculation part and also in finding the factors of the integers.
One can easily find the factors of any integer by expressing that integer as the product of the prime numbers.
 The another way to find L.C.M of $80$, $85$ and $90$ is-
Firstly, we will find the L.C.M of $80$, $85$,
From (2) and (3), $LCM\left( {80,85} \right) = 2 \times 2 \times 2 \times 2 \times 5 \times 17 = 1360$ -(6)
Now to find $LCM\left( {80,85,90} \right)$, we will find the L.C.M of $1360$ and $90$.
So $LCM\left( {80,85,90} \right) = LCM\left( {1360,90} \right)$
Now from (4) and (6), we have $LCM\left( {80,85,90} \right) = LCM\left( {1360,90} \right) = 12240$
Hence this is another way to solve the question.