Question

Three numbers are in the ratio $3:4:5$ and their L.C.M is $2400$. What is their H.C.F?
A. $200$
B. $40$
C. $80$
D. $120$

Answer 1

Hint: Here we need to let the numbers which are in ratio $3:4:5$ as $3x,4x,5x$ and then their LCM comes out to be $60x$. Now we are given the LCM as $2400$ so we can write
$
  60x = 2400 \\
\Rightarrow x = 40 \\
 $
So we get the three numbers as $120,160,200$ and now we need to find the highest common factor of these numbers which will be $40$

Complete step-by-step answer:
Here we are given three numbers are in the ratio $3:4:5$
So let us suppose the numbers to be $3x,4x,5x$
Now we are given their LCM in the numerical value which is $2400$ so we can find the LCM of the three numbers in the term of $x$ and then equate it to $2400$ and get its value.
We know that LCM is the least common multiple of the numbers which are given that is $3x,4x,5x$
We need to find the minimum number which is divisible by $3x,4x,5x$
We know that $3,4,5$ are all prime numbers and in the prime numbers the LCM is simply the multiplication of the numbers
So we get that LCM of $3,4,5$ is $(3)(4)(5) = 60$
As $x$ is common to all so we can say that LCM of $3x,4x,5x$ is $60x$
We are given the LCM of these three numbers as $2400$
So we can say that
$
  60x = 2400 \\
\Rightarrow x = \dfrac{{2400}}{{60}} = 40 \\
$
So we can say that the three numbers are
$
  3x = 3(40) = 120 \\
 4x = 4(40) = 160 \\
  5x = 5(40) = 200 \\
$
Now we need to know what HCF means. It stands for Highest Common Factor which means the highest number that divides all these three numbers.
Here it is clear from the value of
$
  3x = 3(40) = 120 \\
  4x = 4(40) = 160 \\
  5x = 5(40) = 200 \\
$
That $40$ is the highest factor of these three numbers as $3,4,5$ are all prime numbers.
So we get that HCF$ = 40$
Hence B is the correct option.

Note: Here we need to know the meaning of $HCF{\text{ and }}LCM$
HCF stands for Highest Common Factor which means the highest number that divides all the given numbers.
LCM is the Least Common Multiple which means the minimum number that can be divided completely by all the given numbers whose LCM is to be found.