Question

What is the greatest common factor for 36 and 45?

Answer 1

Hint: To obtain the greatest common factor we will use prime factorization method. Firstly we will find the prime factors of the two numbers given then we will add the power of common factors in each of them. Then we will check which all factors are common in both the numbers and finally we will multiply the common factors and get the desired answer.

Complete step by step solution:
The two numbers given are:
$36,45$
Now we will find prime factor of both the number separately as:
For number 36 we get,
$\begin{align}
  & 2\left| \!{\underline {\,
  36 \,}} \right. \\
 & 2\left| \!{\underline {\,
  18 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & 1 \\
\end{align}$
So the prime factors of 36 are:
$36=2\times 2\times 3\times 3$
On simplifying more we can write the above equation as:
$36={{2}^{2}}\times {{3}^{2}}$…..$\left( 1 \right)$
Next, for number 45 we will get,
$\begin{align}
  & 3\left| \!{\underline {\,
  45 \,}} \right. \\
 & 3\left| \!{\underline {\,
  15 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & 1 \\
\end{align}$
So the prime factors of 45 are:
$45=3\times 3\times 5$
On simplifying more we can write above equation as:
$45={{3}^{2}}\times 5$…..$\left( 2 \right)$
On comparing equation (1) and (2) we get common factors as:
Common factor $={{3}^{2}}$
Therefore,
Least common multiple $=9$
Hence the greatest common factor of 36 and 45 is 9.

Note: Greatest common factor also known as H.C.F is the largest factor of given numbers. It is generally used to simplify a fraction value when we have to simplify the denominator and numerator to the smallest number possible. Another way to find the H.C.F is by finding factors of both the numbers then take out the common factors and find the highest factor among them as follows:
\[36=\left\{ 1,2,3,4,6,9,12,18,36 \right\}\]
$45=\left\{ 1,3,5,9,15,45 \right\}$
The common factors in above two values are:
\[\left\{ 1,3,9 \right\}\]
The Highest among them is 9.