Question

Find the HCF of $ 34,102 $.

Answer 1

Hint: Here in this question, we have to find the HCF (Highest common factor), it is the largest or greatest factor common among the two or more given numbers. To find the HCF, we must have to find out its prime factors. Prime factorization is the process of finding which prime numbers can be multiplied together to make the original number.

Complete step by step solution:
The largest positive integer which divides two or more integers without any remainder is called Highest Common Factor (HCF) or Greatest Common Divisor or Greatest Common Factor (GCF).
To find HCF, we have two important methods which are the Prime factorisation method and division method.
Now find the HCF of $ 34,102 $ by Prime factorisation method.
HCF of 34 is:

2	34
17	17
	1

Prime factorization of $ 34 = 2 \times 17{\text{ }}.....{\text{ (a)}} $

2	102
3	51
17	17
	1

Prime factorization of $ 102 = 2 \times 3 \times 17{\text{ }}.....{\text{ (b)}} $
Rewrite equations (a) and (b)
$ 34 = 2 \times 17{\text{ }} $
$ 102 = 2 \times 3 \times 17 $
From the above two equations, we can observe that numbers 2 and 17 are common in both the equations.
To get the HCF (Highest common factor), write as the product of the common factors in case of more than one common factor.
Therefore, HCF of $ 34 $ and $ 102 $ is
 $ \Rightarrow 2 \times 17 $
 $ \Rightarrow 34 $

Note: To solve these types of sums, one should be clear about the concept of HCF and LCM and the prime numbers. HCF is the highest or greatest common multiple whereas the LCM is the least common multiple or least common divisor in two or more given numbers. Prime numbers are the numbers greater than $ 1 $ and which are not the product of any two smaller natural numbers. For Example: $ 2,{\text{ 3, 5, 7,}}...... $ $ 2 $ is the prime number as it can have only $ 1 $ factor. Factors are the number $ 1 $ and the number itself. Also, remember that we get the prime factorization of any composite number.