Question

If LCM of three numbers is 12 and numbers are in the ratio 1:2:3, then the numbers are
A.2, 6, 12
B.4, 8, 12
C.5, 10, 15
D.2, 4, 6

Answer 1

Hint: Firstly, let x be the common factor of the numbers which are in the ratio \[1:2:3\] .
So, the numbers will be x, 2x and 3x.
Hen, find the LCM of the numbers x, 2x and 3x and compare it with the LCM given in the question. Thus, find x.
Finally, after finding x, put the value of x in the numbers and find the numbers.

Complete step-by-step answer:
Let x be the common factor to the numbers.
Here, it is given that the numbers are in the ratio of \[1:2:3\] .
So, the numbers will be x, 2x and 3x.
Now, x can be written as $1 \times x$ , 2x can be written as $2 \times x$ and 3x can be written as $3 \times x$ .
Thus, we get the LCM of x, 2x and 3x as \[LCM\left( {x,2x,3x} \right) = 1 \times 2 \times 3 \times x = 6x\]
Hence, LCM of numbers x, 2x and 3x is 6x.
Also, it is given that in the question, LCM of numbers is 12.
 $
  \Rightarrow 6x = 12 \\
  \Rightarrow x = \dfrac{{12}}{6} \\
  \Rightarrow x = 2 \\
 $
Thus, we get the numbers $1 \times x = 1 \times 2 = 2$ , $2 \times x = 2 \times 2 = 4$ and $3 \times x = 3 \times 2 = 6$ .
So, option (D) is correct.

Note: LCM of numbers:
Least Common Multiple (LCM) of any two numbers, say x and y, is the smallest positive integer that is divisible by both x and y. It is usually denoted by LCM(x, y).
For example, the given two numbers are 2 and 3. So, the LCM of the numbers 2 and 3 will be $2 \times 3 = 6$ . Thus, LCM of the numbers 2 and 3 is 6, which is divisible by both 2 and 3.