Question

Find the HCF and LCM of $117, 221$

Answer 1

Hint:
Here, we will find the factors of the two numbers by using the prime factorization method. Then we will multiply the factors which are common in both the numbers to get the required H.C.F. of the two numbers. We will then multiply all the factors with greatest power to find the L.C.M. The Prime Factorization is a method of finding the factors of the given numbers.

Complete step by step solution:
We are given with the numbers 117, 221.
Now, we will find the factors of all the numbers using the method of prime factorization
Now, we are finding the factors of 117 using prime factorization.
$\begin{array}{*{20}{l}}
  3| {117} \\
\hline
  3| {39} \\
\hline
  {13}| {13} \\
\hline
  {}| 1
\end{array}$
Thus, 117 can be represented as:
$117 = {3^2} \times 13$
Now, we will find the factors of 221 using prime factorization.
$\begin{array}{*{20}{l}}
  {13}| {221} \\
\hline
  {17}| {17} \\
\hline
  {}| 1
\end{array}$
Thus, 221 can be represented as:
$221 = 13 \times 17$
Thus the factors of all the numbers are represented with the same bases as
$117 = {3^2} \times 13 \\
  221 = 13 \times 17 \\ $
Now, we will find the HCF for the given numbers from the factors.
Highest common factor is a factor which is common for all the factors.
As only 13 is the common factor between the two number, so
$HCF\left( {117,221} \right) = 13$
Now, we will find the LCM for the given numbers from the factors.
Least Common Multiple is a multiple which is divisible by all the numbers.
Thus, we get
$LCM\left( {117,221} \right) = {3^2} \times {13^1} \times 17$
Applying the exponent on terms, we get
$ \Rightarrow LCM\left( {117,221} \right) = 9 \times 13 \times 17$
Multiplying the terms, we get
$ \Rightarrow LCM\left( {117,221} \right) = 1989$

Therefore, the HCF of 117 and 221 is 13 and the LCM of 117 and 221 is 1989.

Note:
The Highest Common Factor (H.C.F) of two numbers is defined as the greatest number which divides exactly both the numbers. The Least Common Multiple (L.C.M) of two numbers is defined as the smallest number which is divisible by both the numbers. HCF can be found by multiplying the factors with the least exponent common for all the factors and LCM can be found by multiplying the factors with the highest exponent from all the factors. Common Factor is the factor that is common to all the numbers whereas the prime factors are the factors, which is the product of the powers of the prime numbers.