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

Last updated date: 14th Jun 2024
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Hint: LCM is elaborated as Least Common Multiple, which is the lowest common factor among the integers. To find the LCM of the given numbers, which is the lowest common number, which is divisible by all numbers for which we are finding the LCM, the method includes basic factorization of the numbers to find factors that are multiplied together to form a number. If the all given numbers are \[0\]for which LCM is being calculated, their LCM will also be\[0\]. First, find the factors of the number that are the number when multiplied together gives the original number.
In this question first, assume the common factor to be\[x\], then find the numbers by equating it to the given LCM.

Complete step by step solution: Let us assume the common factor to the number to be\[x\]
Since the number whose LCM is given in the ratio of 1:2:3 hence we can write the number to be $x,2x,3x$.
Now find the factor of each number as:
  1x = 1 \times x \\
  2x = 2 \times x \\
  3x = 3 \times x \\
So the LCM of the three numbers will be:
\[LCM\left( {1x,2x,3x} \right) = 1 \times 2 \times 3 \times x = 6x\]
Hence the LCM of numbers \[1x,2x,3x\] is \[6x\].
Also given the LCM of the original numbers is 12, now equate the LCM as
  6x = 12 \\
  x = \dfrac{{12}}{6} \\
   = 2 \\
Hence the value of\[x = 2\], now find the numbers by putting the value of $x$ in the ratios as:
  1x = 1 \times 2 = 2 \\
  2x = 2 \times 2 = 4 \\
  3x = 3 \times 2 = 6 \\
Hence the numbers whose LCM is 12 are 2,4,6
Option D is correct.

Note: LCM of given numbers is exactly divisible by each of the numbers. During the LCM calculation, students must know the tables of various numbers, and they have to perform the operations step by step. Lastly, they have to multiply all the numbers by which they are dividing the given set of numbers. The least common multiple, lowest common multiple, or smallest common multiple 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.