Question

The LCM of 54, 90 and a third number is 1890 and their HCF is 18. The third number is
(a) 36
(b) 180
(c) 126
(d) 108

Answer 1

Hint: We will first assume the third number as $18x$ because we are given that the HCF is 18. So, one of the factors will be 18. Then, we will write down 54, 90 and $18x$ in its factorization form with one factor as 18. We then have to multiply all the factors listed equate it to 1890 (we will multiply 18 only once). Then, we have to find the value of x from the equation formed and substitute it in 18x.

Complete step by step solution:
We are given that LCM of 54, 90 and a third number is 1890 and their HCF is 18. We have to find the third number. Let us consider the third number as $18x$ because we are given that the Highest Common Factor (HCF) is 18. So, one of the factors will be 18. Now, let us write the 54, 90 and $18x$ in factorization form.
$\begin{align}
  & \Rightarrow 54=18\times 3 \\
 & \Rightarrow 90=18\times 5 \\
 & \Rightarrow 18x=18\times x \\
\end{align}$
We are given that the LCM of 54, 90 and $18x$ is 1890. We have to multiply all the factors listed above and equate it to 1890.
$\Rightarrow 18\times 5\times 3\times x=1890$
Let us find the value of x.
$\Rightarrow 270x=1890$
Let us take 270 to the RHS.
$\Rightarrow x=\dfrac{1890}{270}$
We have to cancel the zeroes from the numerator and denominator.
$\Rightarrow x=\dfrac{189\require{cancel}\cancel{0}}{27\require{cancel}\cancel{0}}$
We can write result of the above simplification is
$\Rightarrow x=\dfrac{189}{27}$
Now, let us divide 189 by 27.
$\Rightarrow x=7$
Therefore, we can find the third number by substituting the above result in 18x.
$\begin{align}
  & \Rightarrow 18x=18\times 7 \\
 & =126 \\
\end{align}$

So, the correct answer is “Option c”.

Note: Students have a chance of making mistakes when writing the factors of 54, 90 and 18x by not writing the factors in terms of their HCF, that is, 18. We know that LCM means Least Common Multiple. So, in order to find the LCM of three numbers, we will multiply their common factors. This is why we multiplied the listed factors and equated it to 1890.