Question

Find the HCF and LCM of $ 625,1125 $ and $ 2125 $ using the fundamental theorem of arithmetic method.
a) $ 95625 $
b) $ 95425 $
c) $ 98621 $
d) $ 92536 $

Answer 1

Hint: Fundamental theorem of arithmetic​is the unique prime factorization method. Here we find out HCF is the highest common factor and another thing we have to find out is the LCM that is the lowest common factor. So, we will first take the lowest number that is divisible by each of three numbers. After that the next number that is divisible by at least two numbers and so on.

Complete step-by-step answer:
We are given three numbers that is $ 625,1125 $ and $ 2125 $ respectively
We are asked to find the solution using prime factorization method
So, let’s do it one by one
First, we have $ 625 $
 $ \Rightarrow 625 = 5 \times 5 \times 5 \times 5 = {5^4} $
Now let’s take another number
We have $ 1125 = 3 \times 3 \times 5 \times 5 \times 5 = {3^2} \times {5^3} $
Let’s take the last number which is
 $ 2125 = 5 \times 5 \times 5 \times 17 = {5^3} \times 17 $
Let’s find out HCF
So we will find the common factors then we will mark out the highest among them
 $ \Rightarrow 625 = 5 \times 5 \times 5 \times 5 = {5^4} $
 $ \Rightarrow 1125 = 3 \times 3 \times 5 \times 5 \times 5 = {3^2} \times {5^3} $
 $ \therefore 2125 = 5 \times 5 \times 5 \times 17 = {5^3} \times 17 $
Clearly, we have common factor as $ {5^3} = 125 $
Hence HCF is found to be 125
Now let’s find out LCM
And LCM is lowest common multiple
 $ \Rightarrow 625 = 5 \times 5 \times 5 \times 5 = {5^4} $
 $ \Rightarrow 1125 = 3 \times 3 \times 5 \times 5 \times 5 = {3^2} \times {5^3} $
 $ \Rightarrow 2125 = 5 \times 5 \times 5 \times 17 = {5^3} \times 17 $
Clearly from this we have LCM as $ {5^4} \times {3^2} \times 17 $
This number is the multiple for all the three numbers
Hence LCM is $ {5^4} \times {3^2} \times 17 = 95625 $
Hence, the HCF of above numbers is $ 125 $ and LCM is $ 95625 $ respectively.
The option we have is $ 95625 $ as option ’a’. So, we can choose it as the correct one
So, the correct answer is “Option A”.

Note: While calculating the LCM if there is no common factor then we can directly multiply each number to get the result. If the prime factor of two numbers don’t have anything as common then HCF can’t be calculated. Such numbers are called co-prime numbers.