Question

Find the HCF of the following numbers:
(a) \[18,48\]
(b) \[30,42\]
(c) \[18,60\]
(d) \[27,63\]
(e) \[36,84\]
(f) \[34,102\]
(g) \[70,105,175\]
(h) \[91,112,49\]
(i) \[18,54,81\]
(j) \[12,45,75\]

Answer 1

Hint:In order to determine the HCF of the given numbers. The highest common factor (HCF) of two numbers is the number that is present in both of them. Let's talk about what HCF is before we start looking for it. The highest common factor, or HCF, is the factor of any two or more numbers that have a common factor. It's also known as the greatest common factor (GCF) or greatest common divisor (GCD). So, for solving the above given numbers we are using the Prime Factorization method to solve it.

Complete step by step answer:
(a) In this problem, we have to find the HCF of the following numbers
\[18,48\]
Using a prime factorization method,
The prime factors of \[18\] and \[48\]can be written as
\[18 = 2 \times 3 \times 3\]
\[\Rightarrow 48 = 2 \times 2 \times 2 \times 2 \times 3\]
The common factors of \[18\] and \[48\] are \[2\], \[3\]. Thus, HCF of \[18,48\] is \[2 \times 3 = 6\]

(b) \[30,42\]
Using the prime factorization method,
The prime factors of \[30\] and \[42\]can be written as
\[30 = 2 \times 3 \times 5\]
\[\Rightarrow 42 = 2 \times 3 \times 7\]
The common factors of \[30\]and \[42\] are \[2\],\[3\]. Thus, HCF of \[30,42\] is \[2 \times 3 = 6\]

(c) \[18,60\]
Using the prime factorization method,
The Prime factors of \[18\] and \[60\]can be written as
\[18 = 2 \times 3 \times 3\]
\[\Rightarrow 60 = 2 \times 2 \times 3 \times 5\]
The common factors of \[18\] and \[60\]are \[2\], \[3\]. Thus, the HCF of \[18,60\] is \[2 \times 3 = 6\].

(d) \[27,63\]
Using the prime factorization method,
The Prime factors of \[27\]and \[63\] can be written as
\[27 = 3 \times 3 \times 3\]
\[\Rightarrow 63 = 3 \times 3 \times 7\]
The common factor of \[27\]and \[63\] is\[3\] , \[3\]. Thus, the HCF of \[27,63\] is \[3 \times 3 = 9\].

(e) \[36,84\]
Using the prime factorization method,
The Prime factors of \[36\] and \[84\]can be written as
\[36 = 2 \times 2 \times 3 \times 3\]
\[\Rightarrow 84 = 2 \times 2 \times 3 \times 7\]
The common factor of \[36\] and \[84\] is \[2\],\[2\],\[3\]. Thus, the HCF of \[36,84\]is \[2 \times 2 \times 3 = 12\]

(f) \[34,102\]
Using the prime factorization method,
The Prime factors of \[34\] and \[102\]can be written as
\[34 = 2 \times 17\]
\[\Rightarrow 102 = 2 \times 3 \times 17\]
The common factor of \[34\] and \[102\]is \[2\],\[17\]. Thus, the HCF of \[34,102\]is \[2 \times 17 = 34\]

(g) \[70,105,175\]
Using the prime factorization method,
The prime factors of \[70\],\[105\]and \[175\]can be written as
\[70 = 2 \times 5 \times 7\]
\[\Rightarrow 105 = 3 \times 5 \times 7\]
\[\Rightarrow 175 = 5 \times 5 \times 7\]
The common factors of \[70\],\[105\] and \[175\] are \[5\],\[7\]. Thus, the HCF of \[70,105,175\] is \[5 \times 7 = 35\].

(h) \[91,112,49\]
Using the prime factorization method,
The prime factors of \[91\],\[112\] and \[49\] can be written as
\[91 = 7 \times 13\]
\[\Rightarrow 112 = 2 \times 2 \times 2 \times 2 \times 7\]
\[\Rightarrow 49 = 7 \times 7\]
The common factor of \[91\],\[112\]and \[49\]is \[7\]. Thus, the HCF of \[91,112,49\]is \[7\]

(i) \[18,54,81\]
Using the prime factorization method,
The prime factors of \[18\],\[54\] and \[81\] can be written as
\[18 = 2 \times 3 \times 3\]
\[\Rightarrow 54 = 2 \times 3 \times 3 \times 3\]
\[\Rightarrow 81 = 3 \times 3 \times 3 \times 3\]
The common factors of \[18\],\[54\] and \[81\] are \[3\] and \[3\]. Thus, the HCF of \[18,54,81\] is \[3 \times 3 = 9\]

(j) \[12,45,75\]
Using the prime factorization method,
The prime factors of \[12\],\[45\] and \[75\] can be written as
\[12 = 2 \times 2 \times 3\]
\[\Rightarrow 45 = 5 \times 3 \times 3\]
\[\Rightarrow 75 = 5 \times 5 \times 3\]
The common factor of \[12\],\[45\] and \[75\]is \[3\]. Thus, the HCF of \[12,45,75\] is \[3\].

Note:The product of any two provided natural numbers' LCM and HCF is the same as the product of the supplied numbers. Co-prime numbers have an HCF of \[1]. As a result, the product of the numbers is equal to the LCM of the specified co-prime numbers.