Question

The G.C.D. of 1.08, 0.36 and 0.9 is
(a) 0.03
(b) 0.9
(c) 0.18
(d) 0.108

Answer 1

Hint: At first change decimals into fractions such that denominator is same. We can do this by making the denominator as 100 for all terms. Then find G.C.D. of numerator and then divide it by L.C.M of denominator. G.C.D will be the product of all factors which ae common to all terms.

Complete step-by-step answer:
In the question, we have to find a G.C.D or greater common divisor of 1.08, 0.36, 0.90.
At first we briefly understand G.C.D or greatest common divisor.
In Mathematics, the greatest common divisor or G.C.D. of two or more integers which are not at all zero, the largest positive integer that divides each of the integers. For example, if g.c.d of 8 and 12 is 4.
In the name greatest common divisor, the adjective ‘greatest’ can be replaced by ‘highest’ and the word ‘divisor’ that can be replaced by ‘factor’, so that the names include highest common factor (hcf).
 Let’s find the greatest common divisor of 54, 24. So, we will first factorize 54, 24 and write it in the form of a product of prime factors.
At first we will do 54.
$\begin{align}
  & 2\left| \!{\underline {\,
  54 \,}} \right. \\
 & 3\left| \!{\underline {\,
  27 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & \text{ 3} \\
\end{align}$
So, we have got $\text{54=2}\times {{\text{3}}^{3}}$
Now we will do for 24.
$\begin{align}
  & 2\left| \!{\underline {\,
  24 \,}} \right. \\
 & 2\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & \text{ 3} \\
\end{align}$
So, we get that $\text{24=}{{\text{2}}^{3}}\times 3$
Hence, we can write $\text{54=2}\times {{\text{3}}^{3}}$ and $\text{24=}{{\text{2}}^{3}}\times 3$ .
The common factors are $2\times 3$ so the g.c.d is 6.
If the fractions are given then we will find g.c.d by taking the value of g.c.d. of numerators and dividing it by l.c.m of denominator.
Now, coming to our solution. The decimals given are 1.08, 0.36 and 0.90.
So, we write it in fraction form as
$\dfrac{108}{100},\dfrac{36}{100},\dfrac{90}{100}$
For finding the hcf of the above fractions, we have to calculate hcf of 108, 36 and 90 and then divide it by 100 because their denominator is the same.
Now, we will factorize 108 first.
$\begin{align}
  & 2\left| \!{\underline {\,
  108 \,}} \right. \\
 & 2\left| \!{\underline {\,
  54 \,}} \right. \\
 & 3\left| \!{\underline {\,
  27 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & \text{ 3} \\
\end{align}$
So, we have got $\text{108=}{{\text{2}}^{2}}\times {{3}^{3}}$
Now, we will do for 36
$\begin{align}
  & 2\left| \!{\underline {\,
  36 \,}} \right. \\
 & 2\left| \!{\underline {\,
  18 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & \text{ 3} \\
\end{align}$
So, we get that $\text{36=}{{\text{2}}^{2}}\times {{3}^{2}}$
Now, for the number 90.
$\begin{align}
  & 2\left| \!{\underline {\,
  90 \,}} \right. \\
 & 3\left| \!{\underline {\,
  45 \,}} \right. \\
 & 3\left| \!{\underline {\,
  15 \,}} \right. \\
 & \text{ 5} \\
\end{align}$
So, we have got $\text{90=2}\times {{3}^{2}}\times 5$
Hence, we can write
$\text{108=}{{\text{2}}^{2}}\times {{3}^{3}}$ and $\text{36=}{{\text{2}}^{2}}\times {{3}^{2}}$ and $\text{90=2}\times {{3}^{2}}\times 5$
The common factors are $\text{2}\times {{3}^{2}}$ so, the g.c.d is 18.
So the g.c.d of 1.08, 0.36, 0.9 is $\dfrac{18}{100}$ or 0.18.
So, the correct option is ‘c’.

So, the correct answer is “Option C”.

Note: For finding G.C.D of fractions we have to divide g.c.d of numerator by l.c.m of denominator. If we have to find l.c.m of fractions, then we have to divide lcm of numerator by g.c.d of denominator. Make sure that the denominator of the terms are the same, the mistake that can happen is students might consider the denominator of 0.9 as 10.