Question

Three numbers are in the ratio $ 3:4:5 $ . Their LCM is 2400. Find the HCF of those numbers.

Answer 1

Hint: We first assume the ratio constant for the numbers. We form the numbers and using the simultaneous factorisation, we find the HCF and LCM. We solve the equation and find the value of the variable which gives the HCF.

Complete step-by-step answer:
Let us assume the ratio constant as $ x $ . Three numbers are in the ratio $ 3:4:5 $ .
Therefore, the numbers are $ 3x,4x,5x $ . Now we find the HCF and LCM of the numbers.
We use the simultaneous factorisation to find the HCF and LCM of the numbers $ 3x,4x,5x $ .
We have to divide both of them with possible primes which can divide both of them.
\[\begin{align}
  & x\left| \!{\underline {\,
  3x,4x,5x \,}} \right. \\
 & 1\left| \!{\underline {\,
  3,4,5 \,}} \right. \\
\end{align}\]
The LCM is $ 3\times 4\times 5\times x=60x $ and the HCF is $ x $ .
It is given that LCM is 2400. So, $ 60x=2400 $ . We simplify to find the value of $ x $ .
 $ \begin{align}
  & 60x=2400 \\
 & \Rightarrow x=\dfrac{2400}{60}=40 \\
\end{align} $
The value $ x $ represents the HCF. Therefore, the HCF is 40.
So, the correct answer is “40”.

Note: We need to remember that the GCF has to be only one number. It is the greatest possible divisor of all the given numbers. If the given numbers are prime numbers, then the GCD of those numbers will always be 1. Therefore, if for numbers $ x $ and $ y $ , the GCD is $ a $ then the GCD of the numbers $ \dfrac{x}{a} $ and $ \dfrac{y}{a} $ will be 1.