Question

Find the HCF of the following numbers, using the prime factorization method: $144$ ,$252$ ,$630$

Answer 1

Hint: To solve this problem, first we will find the prime factors of all the three given numbers. Then will observe the resultant factors and note down the common factors from all the three numbers. Here, we will find factors by division method.

Complete step by step solution:
In this prime factorization method, first we will find the factors of the given numbers by a division method where you have to start dividing the number with least prime numbers till the further division is not possible. Gradually try to divide the number first by $2$ then $3,5,7,11$ and so on...
\[
  \underline 2 |\underline {144} \\
  \underline 2 |\underline {72} \\
  \underline 2 |\underline {36} \\
  \underline 2 |\underline {18} \\
  \underline 3 |\underline 9 \\
  \underline 3 |\underline 3 \\
  {\text{ | }}1\; \\
 \]
Therefore, the factors of $144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3{\text{ }}.....{\text{ (a)}}$
 \[
  \underline 2 |\underline {252} \\
  \underline 2 |\underline {126} \\
  \underline 3 |\underline {36} \\
  \underline 3 |\underline {21} \\
  \underline 7 |\underline 7 \\
  {\text{ | }}1\; \\
 \]
Therefore, the factors of $252 = 2 \times 2 \times 3 \times 3 \times 7{\text{ }}.....{\text{ (b)}}$
\[
  \underline 2 |\underline {630} \\
  \underline 3 |\underline {315} \\
  \underline 3 |\underline {15} \\
  \underline 5 |\underline 5 \\
  {\text{ | }}1\; \\
 \]
Therefore, the factors of $630 = 2 \times 3 \times 3 \times 5\,{\text{ }}.....{\text{ (c)}}$
Now, re-writing all the three numbers along with the factors –
$144 = \underline 2 \times 2 \times 2 \times 2 \times \underline 3 \times \underline 3 $
$252 = \underline 2 \times 2 \times \underline 3 \times \underline 3 \times 7$
$630 = \underline 2 \times \underline 3 \times \underline 3 \times 5$
Underline the common factors from all the three numbers, and write as the product of the common factors in case of more than one common factor.
Therefore, HCF = $ 2 \times 3 \times 3$
HCF = $18$
Hence, the required answer - HCF of $144,252,630$ , by using the prime factorization method is $18$

Note: Remember the multiples of the numbers till at least twenty for an accurate and efficient answer. Remember the basic difference between the HCF (Highest common factor) and LCM (Least common multiple) to solve these types of sums and apply accordingly. HCF is the greatest or the largest common factor between two or more given numbers whereas the LCM is the least or the smallest number with which the given numbers are exactly divisible.