Question

Find the HCF of the following numbers. \[70,105,175\].

Answer 1

Hint: Here, we will find the factors of the two numbers by using the prime factorization method. Then we will multiply the factors which are common in both the numbers to get the required highest common factor (H.C.F.) of the two numbers. Prime Factorization is a method of finding the factors of the given numbers in terms of prime numbers.

Complete Step by Step Solution:
We are given the numbers 70, 105 and 175
Now, we will find the factors of all the numbers using the method of prime factorization
First, we will find the factors of 70 using prime factorization.
Now as 70 is an even number, so we will divide it by the least prime number 2. Therefore, we get
\[70 \div 2 = 35\]
Here, we cannot divide 35 by 3 as 3 is not a factor of 35.
Now dividing the 35 by next least prime number 5, we get
\[35 \div 5 = 7\]
Now as we have obtained our quotient as prime number, we will not factorize it further.
Thus, the factors of 70 are 2, 5 and 7 and can be written as:
\[70 = 2 \times 5 \times 7\]
We will now find the factors of 105 using prime factorization, we get
Now as 105 is an odd number, so we will divide it by least odd prime number 3.
\[105 \div 3 = 35\]
Now dividing the 35 by next least prime number 5, we get
\[35 \div 5 = 7\]
Now as we have obtained our quotient as prime number, we will not factorize it further.
Thus, the factors of 150 are 3, 5 and 7 and can be written as:
\[105 = 3 \times 5 \times 7\]
Now, we will find the factors of 175 using prime factorization.
Now as 175 is an odd number, so we will divide it by least prime number 5.
\[175 \div 5 = 35\]
Now dividing the 35 by 5, we get
\[35 \div 5 = 7\]
Now as we have obtained our quotient as prime number, we will not factorize it further.
Thus, the factors of 175 are 5, 5 and 7 and can be written as:
\[175 = 5 \times 5 \times 7\]
Thus the factors of all the numbers are represented with the same bases as:
\[\begin{array}{l}70 = {2^1} \times {3^0} \times {5^1} \times {7^1}\\105 = {2^0} \times {3^1} \times {5^1} \times {7^1}\\175 = {2^0} \times {3^0} \times {5^2} \times {7^1}\end{array}\]
Now, we will find the HCF for the given numbers from the factors.
Highest common factor is a factor which is common for all the factors.
Thus, we get
\[HCF\left( {70,105,175} \right) = {2^0} \times {3^0} \times {5^1} \times {7^1}\]
Applying the exponent, we get
\[ \Rightarrow HCF\left( {70,105,175} \right) = 1 \times 1 \times 5 \times 7\]
Multiplying the terms, we get
\[ \Rightarrow HCF\left( {70,105,175} \right) = 35\]

Therefore, the HCF of 70, 105 and 175 is 35.

Note:
The Highest Common Factor (H.C.F) of two numbers is defined as the greatest number which divides exactly both the numbers. In order to find the HCF of two numbers, it is important to express those numbers as a product of their prime factors. Prime factors are those factors which are greater than 1 and have only two factors, i.e. factor 1 and the prime number itself. So, in order to express the given number as a product of its prime factors, we are required to do the prime factorization of the given number.