Question

# The ratio of two numbers is 2:3. If their LCM is 150, find the numbers.(A) 30, 45 (B) 48, 72 (C) 50, 75 (D) 26, 39

Hint:
LCM stands for least common multiple or lowest common multiple. LCM of two integers a and b, usually denoted by lcm (a, b), is the smallest positive integer that is divisible by both a and b. In simple words it is the smallest number that they have in common as a multiple.

Complete step by step solution:
Here the ratio of the two numbers is given as 2:3.
The two numbers can be expressed as 2x and 3x, where x is an integer.
Since 150 is the LCM of 2x and 3x, it is a common multiple of the two numbers, so x must divide 150,
$\begin{gathered} Lcm(2x,3x) = 6x = 150 \\ \Rightarrow 6x = 150 \\ \Rightarrow x = 25 \\ \end{gathered}$
Putting the value of x in the two numbers, we get, $2x = 2 \times 25 = 50$ and $3x = 3 \times 25 = 75$.
Therefore, the two numbers are 50 and 75. Thus $lcm(50,75) = 150$.
The correct option is (C).

Note:
For the lowest common multiple to be between 2x and 3x, there must be one 2, one 3 and two 5’s between them. This is satisfied when x is 25. If we use prime factorisation method then:
Prime factorization of 50 is: 2X5X5
Prime factorization of 75 is: 3X5X5
Eliminating duplicates factors and multiplying the remaining we get 150. Thus, verifying our answer above. These questions usually pop up in Olympiads and other exams, to solve such questions, try to learn how to find LCM and HCF effectively. Also memorise the relation between LCM and HCF.