Question

Find the LCM of the following numbers.
12 and 5
Observe a common property in the obtained LCMs. Is LCM the product of two numbers?

Answer 1

Hint: We are given two numbers 12 and 5 and we are asked to find the LCM of these numbers. We will carry out the factorization and we will have the LCM of the numbers. If we observe the given numbers, they have only 1 as the common factor between them and such numbers are called co – prime numbers. In case of co – prime numbers, the LCM of those numbers is equal to the product of those numbers. Hence, we will have the required answer.

Complete step-by-step solution:
According to the given question, we are given two numbers, which are, 12 and 5 and we are asked to find the LCM of the given numbers.
The numbers we have are,
12 and 5
We will now be calculating the LCM of 12 and 5 using the division method. So, we have,
\[\begin{align}
  & 2\left| \!{\underline {\,
  12,5 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6,5 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3,5 \,}} \right. \\
 & 5\left| \!{\underline {\,
  1,5 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1,1 \,}} \right. \\
\end{align}\]
So, we get the LCM as,
\[LCM=2\times 2\times 3\times 5=60\]
Now, if we look at the factors of the two numbers involved, that is,
\[12=2\times 2\times 3\times 1\]
\[5=5\times 1\]
We can observe that the numbers 12 and 5 have only 1 as the common factor between them and so we can say that these numbers are co – prime numbers. Co – prime numbers are those prime numbers which have 1 as the only common multiple.
And also, if we want to find the LCM of co – prime numbers then the LCM is equal to the product of the co – prime numbers. Like in the above solution, 12 and 5 have only 1 as the common multiple. So, their LCM is equal to the product of 12 and 5, that is, we get,
\[LCM=12\times 5=60\]
Therefore, the LCM of 12 and 5 is 60. And, yes, the LCM is the product of 12 and 5 since 12 and 5 are co-prime numbers.

Note: The LCM of the numbers should not be confused with the HCF of the numbers. LCM refers to the least common multiples in the numbers involved whereas HCF refers to the highest common factor in the numbers involved.
To find the LCM of 12 and 5, we have,
\[\begin{align}
  & 2\left| \!{\underline {\,
  12,5 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6,5 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3,5 \,}} \right. \\
 & 5\left| \!{\underline {\,
  1,5 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1,1 \,}} \right. \\
\end{align}\]
\[LCM=2\times 2\times 3\times 5=60\]
And, the HCF of the two numbers 12 and 5 is,
\[12=2\times 2\times 3\times 1\]
\[5=5\times 1\]
\[HCF(12,5)=1\]