Question

What is the LCM of \[6\] and \[8\]

Answer 1

Hint: In this question we have found the LCM of 6 and 8. The LCM is the least common multiple and it is defined as \[LCM(a,b) = \dfrac{{\left| {a \cdot b} \right|}}{{\gcd (a,b)}}\] , where a, b and c are integers and \[\gcd \]is the greatest common divisor. We can find LCM by division method also. Here we solve by formula and also by division form.

Complete step-by-step answer:
Consider the given numbers 6 and 8.
First, we divide the each by 2 if the number divides by 2 then we write the quotient otherwise we write same number in the next line. Next, we will divide the numbers by 2 and same procedure of writing is carried out. Again, next we divide by 2 and the same as above. Again, next we divide by 3 and the same as above and hence we obtain 1 in the last row. This is end od of the division procedure. We have to divide till we get 1 in the next row.
\[
  2\,\left| \!{\underline {\,
  {6,8} \,}} \right. \\
  2\,\left| \!{\underline {\,
  {3,\,4} \,}} \right. \\
  2\,\left| \!{\underline {\,
  {3,\,2} \,}} \right. \\
  3\,\,\left| \!{\underline {\,
  {3,\,1} \,}} \right. \\
  \,\,\,\,\,1,\,\,1 \\
 \]
Here we will consider the numbers by which we have divided the three numbers.
Now to find LCM of the given numbers we have to multiply the first column numbers that is
\[LCM = 2 \times 2 \times 2 \times 3\]
\[ \Rightarrow LCM = 24\]
Therefore, the LCM of 6 and 8 is 24.
Hence we can determine the LCM by formula method
\[LCM(a,b) = \dfrac{{\left| {a \cdot b} \right|}}{{\gcd (a,b)}}\]
The factors of 6 and 8 is given by
\[ \Rightarrow 6 = 3 \times 2\]
\[ \Rightarrow 8 = 2 \times 2 \times 2 \times 2\]
The common factors of 6 and 8 are 2. Therefore \[\gcd (6,8) = 2\]
On considering the formula we have
 \[ \Rightarrow LCM(6,8) = \dfrac{{\left| {6 \cdot 8} \right|}}{{\gcd (6,8)}}\]
On simplifying we have
\[ \Rightarrow LCM(6,8) = \dfrac{{\left| {48} \right|}}{2}\]
\[ \Rightarrow LCM(6,8) = \dfrac{{48}}{2}\]
\[ \Rightarrow LCM(6,8) = 24\]
Therefore, the LCM of 6 and 8 is 24.
So, the correct answer is “24”.

Note: We must know about the multiplication, division and tables of multiplication to solve the question. We should divide by the number by the least number and hence it is the correct way to solve the problem.