Question

What is the G.C.F and L.C.M of 36 and 45?

Answer 1

Hint: First of all write the given numbers as the product of their prime factors using the prime factorization method. Now, to find the G.C.F of the two numbers, take the product of all the common factors present even if they are repeated. Further, to find the L.C.M, take the product of all the different prime factors along with their highest exponent to get the answer.

Complete step by step solution:
Here we have been asked to find the G.C.F and the L.C.M of the given numbers 36 and 45. Let us use the prime factorization method to get the answers.
(1) In arithmetic and number theory, the G.C.F (greatest common factor) of two or more integers is the greatest integer which divides all the given numbers without leaving any remainder. There are two methods to find the G.C.F but here we will use the prime factorization method to get the answer. So writing the given numbers 36 and 45 as the product of their prime factors we get,
$\Rightarrow 36=2\times 2\times 3\times 3={{2}^{2}}\times {{3}^{2}}$
\[\Rightarrow 45=3\times 3\times 5={{3}^{2}}\times 5\]
Now, the G.C.F will be the product of all the common prime factors present. Clearly we can see that the factor ${{3}^{2}}$ is common, therefore the G.C.F will be given as: -
\[\Rightarrow \] G.C.F = $3\times 3$
\[\therefore \] G.C.F = 9
Hence, the G.C.F of 36 and 45 is a 9.
(2) Now, the least common multiple (L.C.M) of two or more integers is the smallest positive integer that is divisible by each of the given numbers. There are two methods to determine the L.C.M of two or more given numbers. Here, we will use the method of prime factorization.
$\Rightarrow 36=2\times 2\times 3\times 3={{2}^{2}}\times {{3}^{2}}$
\[\Rightarrow 45=3\times 3\times 5={{3}^{2}}\times 5\]
Here, the L.C.M will be the product of the prime factors along with their highest exponent present. Clearly we can see that the highest power of the prime factors 2 is 2, 3 is 2 and 5 is 1. So, we need to multiply ${{2}^{2}}$, ${{3}^{2}}$ and ${{5}^{1}}$ to get the L.C.M.
\[\Rightarrow \] L.C.M = \[{{2}^{2}}\times {{3}^{2}}\times {{5}^{1}}\]
\[\Rightarrow \] L.C.M = \[4\times 9\times 5\]
$\therefore $ L.C.M = 180
Hence, the L.C.M of 36 and 45 is 180.

Note: Note that the G.C.F is also called the H.C.F or highest common factor. There is one more method known as the long division method to get the H.C.F. However, this long division method is generally used when the numbers are large and they are primes in nature. Also we have a different method for calculating the L.C.M. In this method we write multiples of the given numbers and check which multiple is first common in all of them. However, if the numbers are large then we will face some difficulty in writing their multiples.