Question

Find the L.C.M. and H.C.F. of the following
\[\dfrac{3}{4},\dfrac{6}{7},\dfrac{9}{8}\]

Answer 1

Hint: Here, we will use the rules of LCM and HCF of the fraction to find the required value. We will first find the LCM of the numerator and HCF of the denominator of all the given fraction. We will then substitute the obtained values in the formula and find the LCM of the given fractions. Then we will find the HCF of the numerator and LCM of the denominator of all the given fractions and use the formula to find the required HCF.

Complete step-by-step answer:
In order to find L.C.M. and H.C.F. of fractions, we have a special rule.
The rule of finding the Lowest Common Factor or L.C.M. of fractions is L.C.M. of Numerators \[ \div \]H.C.F. of Denominators.
Now, in this question, the given fractions are \[\dfrac{3}{4},\dfrac{6}{7},\dfrac{9}{8}\].
Now, for finding the L.C.M. first of all we will find the L.C.M. of the numerators.
The numerators are: 3, 6, 9
The factors of 3, 6, 9 is:
\[3 = 1 \times 3\]
\[6 = 2 \times 3\]
\[9 = 3 \times 3 = {3^2}\]
Now, for the L.C.M. we take into consideration, all the factors possible and the highest power of the common factor.
Hence, L.C.M. of the numerators \[ = 1 \times 2 \times {3^2} = 2 \times 9 = 18\]
Now, the denominators are: 4, 7, 8
Clearly, there is no factor other than 1 which is common in all these three numbers.
Hence, H.C.F. of the denominators is 1.
Therefore,
L.C.M. of \[\dfrac{3}{4},\dfrac{6}{7},\dfrac{9}{8} = \] L.C.M. of Numerators \[ \div \] H.C.F. of Denominators
\[ \Rightarrow \] L.C.M. of \[\dfrac{3}{4},\dfrac{6}{7},\dfrac{9}{8}\] \[ = \dfrac{{18}}{1} = 18\]
Hence, the required L.C.M. of \[\dfrac{3}{4},\dfrac{6}{7},\dfrac{9}{8}\] is 18.
Now,
The rule of finding the Highest Common Factor or H.C.F. of fractions is:
H.C.F. of Numerators \[ \div \] L.C.M. of Denominators
Now, in this question, the given fractions are \[\dfrac{3}{4},\dfrac{6}{7},\dfrac{9}{8}\].
Now, for finding the H.C.F. first of all we will find the H.C.F. of the numerators.
The numerators are: 3, 6, 9
The factors of 3, 6, 9 is:
\[3 = 1 \times 3\]
\[6 = 2 \times 3\]
\[9 = 3 \times 3 = {3^2}\]
Now, for the H.C.F. we take into consideration, all the lowest common factors possible.
Hence, H.C.F. of the numerators\[ = 3\]
Now, the denominators are: 4, 7, 8
The factors of 4, 7, 8 are:
\[4 = 2 \times 2 = {2^2}\]
\[7 = 1 \times 7\]
\[8 = 2 \times 2 \times 2 = {2^3}\]
 Now, for the L.C.M. we take into consideration, all the factors possible and the highest power of the common factor.
Hence, L.C.M. of the denominators\[ = 1 \times 7 \times {2^3} = 7 \times 8 = 56\]
\[ \Rightarrow \] L.C.M. of 4, 7, 8\[ = 56\]
Therefore,
H.C.F. of \[\dfrac{3}{4},\dfrac{6}{7},\dfrac{9}{8} = \] H.C.F. of Numerators \[ \div \] L.C.M. of Denominators
\[ \Rightarrow \] H.C.F. of \[\dfrac{3}{4},\dfrac{6}{7},\dfrac{9}{8} = \dfrac{3}{{56}}\]
Hence, the required H.C.F. of \[\dfrac{3}{4},\dfrac{6}{7},\dfrac{9}{8}\] is \[\dfrac{3}{{56}}\]
Therefore, the L.C.M. and H.C.F. of \[\dfrac{3}{4},\dfrac{6}{7},\dfrac{9}{8}\] are 18 and \[\dfrac{3}{{56}}\] respectively.

Note: In order to find the L.C.M. or least common multiple of two or more numbers, we take the product of all the possible factors of the given numbers. For the factors which are common, we take their highest power. Whereas, for finding the H.C.F. or Highest Common Factor of two or more numbers we take only those factors which are common in all the given numbers. Also, the least possible power is taken in the case of H.C.F.