Question

Find the greatest common factor of \[65\] and \[25\].

Answer 1

Hint: Here, in the given question, we are asked to find the greatest common factor between two numbers given. Greatest common factor is also known as the highest common factor (H.C.F.). It means the greatest factor available in two or more given natural numbers. Commonly, there are two methods to calculate the greatest common factor: (i) Prime factorization method, and (ii) Long Division Method. We will use the Prime factorization method to solve the given question. At first, we will factorize two given numbers and then, we will pick up the common factors. And then, the product of all common factors will be the desired result.

Complete step-by-step solution:
Given two numbers are \[65\] and \[25\]
Let us first write all the prime factors of two given numbers, \[65\] and \[25\]
\[
  65 = 5 \times 13 \\
  25 = 5 \times 5 \]
Now, the common factor between\[65\] and \[25\] is \[5\]
Hence, the greatest common factor of \[65\] & \[25\] is \[5\].
Additional information: There is one more method available except for the two methods discussed above named Euclidean Algorithm. In this method, to find the GCF, we will firstly divide the larger number by the smaller one. Like in the given question, divide \[65\] by \[25\] to get a quotient of \[2\] and remainder of \[15\]. Then, we divide the smaller number i.e. \[25\] by the remainder from the last step i.e. \[15\]. So, \[25\] divided by \[15\], we get a quotient of \[1\] and remainder \[10\] . Now, again we will divide \[15\] by the last remainder i.e. \[10\] and continue this until we get a remainder \[0\]. And the last number we used to divide will be the answer.

Note: Avoid using long division methods in such types of questions where writing all the prime factors of the given numbers is easier. In case of co-prime numbers, H.C.F will always be equal to 1. H.C.F of two or more numbers, in any case, can never be greater than any of the given numbers. H.C.F. of two natural numbers can never be zero.