Question

The HCF of the fraction $\dfrac{8}{21},\dfrac{12}{35}\text{ and }\dfrac{32}{7}$ is
A.$\dfrac{4}{105}$
B.$\dfrac{192}{7}$
C.$\dfrac{4}{7}$
D.$\dfrac{5}{109}$

Answer 1

Hint:For finding the HCF of the given fractions you need to use the formula $HCF=\dfrac{HCF\text{ of numerator of the fractions}}{\text{LCM of the denominator of the fractions}}$ . So, find the HCF of the numerators and LCM of the denominators, separately using the prime factorisation method and put the values in the formula to get the required answer.

Complete step-by-step answer:
Before proceeding with the solution, let’s understand the concept of prime factorization. A prime number is a number which is not divisible by any other number except 1 and itself. Any number can be expressed as a product of prime numbers. All the prime numbers, which when multiplied, give a product equal to a number (say x) are called the prime factors of the number x. To write the prime factors of a number, we should always start with the smallest prime number, i.e. 2 and check divisibility. If the number is divisible by the prime number, then we write the number as a product of the prime number and another number, which will be the quotient when the given number is divided by the prime number. Then, we take the quotient and repeat the same process. This process is repeated till we are left with 1 as the quotient.
For example: Consider the number 51. It is an even number. So, it is not divisible by 2. The sum of the digits of 51 is 5 + 1 = 6. Hence, 51 is divisible by 3. Now, $51=3\times 17$ . Now, we take 17. We know, 17 is a prime number. Hence, the prime factors of 51 are 3 and 17.
Let us first find the HCF of the numerators. The numerators of the numbers given in the question are 8, 12 and 32. If we see all the three numbers are a multiple of 4 and 4 can be further written as the product of two 2s. Therefore, these numerators can be written as:
$\begin{align}
  & 8=2\times 2\times 2 \\
 & 12=2\times 2\times 3 \\
 & 32=2\times 2\times 2\times 2\times 2 \\
\end{align}$
So, the common prime factors are two 2s. Therefore, the HCF of the numerators of the given fraction is $2\times 2=4$ .
Now, let us find the LCM of the denominators. The denominators of the numbers given in the question are 21, 35 and 7. If we see all the three numbers are a multiple of 7 . Therefore, these denominators can be written as:
\[\begin{align}
  & 21=3\times 7 \\
 & 7=7\times 1 \\
 & 35=5\times 7 \\
\end{align}\]
So, among the factors of the denominators, 7 appears a maximum of 1 time along with 3 and 5 also appearing a maximum of 1 time each. Therefore, the LCM of the denominator is $7\times 3\times 5=105$ .
So, the HCF of $\dfrac{8}{21},\dfrac{12}{35}\text{ and }\dfrac{32}{7}$ is:
$HCF=\dfrac{HCF\text{ of numerator of the fractions}}{\text{LCM of the denominator of the fractions}}=\dfrac{4}{105}$
Hence, the answer to the above question is option (a).

Note: Be careful while finding the prime factors of each number. Also, it is prescribed that you learn the division method of finding the HCF as well, as it might be helpful. If in case you are asked to find the LCM of two fractions you must use the formula $LCM=\dfrac{LCM\text{ of numerator of the fractions}}{HCF\text{ of the denominator of the fractions}}$ .