Question

Find the H.C.F of the following numbers using prime factorization method.
540, 980.

Answer 1

Hint: First, we should know the concept of prime factorization i.e. finding which prime numbers multiply together to make the original number. Also, the concept of highest common factor (HCF) i.e. the greatest number which can divide the given numbers. So, here we will first find out prime factors of 540, 980 and then we find which factors are common in both that will be our answer.

Complete step-by-step answer:
We know the definition of highest common factor (HCF) given as the greatest number which can divide the given numbers. Also, Prime factorization is finding which prime numbers multiply together to make the original number. To understand this, we will take an example: Suppose we have to find factors of number 12. So, first we will see whether the number is divisible by 2 or not. So, we will get $12=2\times 6$. Now, we will take 6 and see whether it is divided by 2 or not. So, we will get $12=2\times 2\times 3$. Now, we know that 3 is a prime number so no need to further solve this. Hence, we get our prime factors 2 and 3.
So, here the prime factors of 540, 980 are as follows:
 $ \begin{align}
  & 2\left| \!{\underline {\,
  540 \,}} \right. \\
 & 3\left| \!{\underline {\,
  270 \,}} \right. \\
 & 3\left| \!{\underline {\,
  90 \,}} \right. \\
 & 3\left| \!{\underline {\,
  30 \,}} \right. \\
 & 2\left| \!{\underline {\,
  10 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
\end{align} $
Factors of 540 are $ 540=2\times 3\times 3\times 3\times 2\times 5 $
 $ \begin{align}
  & 2\left| \!{\underline {\,
  980 \,}} \right. \\
 & 2\left| \!{\underline {\,
  490 \,}} \right. \\
 & 5\left| \!{\underline {\,
  245 \,}} \right. \\
 & 7\left| \!{\underline {\,
  49 \,}} \right. \\
 & 7\left| \!{\underline {\,
  7 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
\end{align} $
Factors of 980 are $ 980=2\times 2\times 5\times 7\times 7 $
Now, if we compare both the factors, we can see that the highest common factor in both is \[2\times 2\times 5=20\] .
Thus, we can say that HCF of 540, 980 is 20.

Note: Be careful in HCF and LCM i.e. least common multiple. Students generally make mistakes in finding HCF. If we find LCM of 540, 980 we will get an answer as 26460 which is totally wrong. So, do understand the concept of HCF and LCM and then solve it accordingly.