Question

What is the LCM of \[40\], \[42\], \[45\] ?

Answer 1

Hint: In this question we have found the LCM of 40, 42 and 45. 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 and b 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 \[40\], \[42\] and \[45\]. Now we use the division method to find the LCM of given numbers. First, we divide each by 2 if the number divides by 2 then we write the quotient otherwise we write the same number in the next line. Next, we will divide the numbers by 2 and the same procedure of writing is carried out. Similarly, next we divide by 5, 2, 3, 3, 7 and the same as above hence we obtain 1 in the last row. This is the end of the division procedure. We have to divide till we get 1 in the next row.

\[
  2\,\left| \!{\underline {\,
  {40,42,45} \,}} \right. \\
  2\,\left| \!{\underline {\,
  {20,\,21,45} \,}} \right. \\
  5\,\left| \!{\underline {\,
  {10,\,21,45} \,}} \right. \\
  2\,\,\left| \!{\underline {\,
  {2,\,21,9} \,}} \right. \\
  3\,\,\left| \!{\underline {\,
  {1,\,21,9} \,}} \right. \\
  3\,\,\left| \!{\underline {\,
  {1,\,7,3} \,}} \right. \\
  7\,\,\left| \!{\underline {\,
  {1,\,7,1} \,}} \right. \\
  \,\,\,\,\,\,1,1,1 \\
 \]
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
\[LCM = 2 \times 2 \times 5 \times 2 \times 3 \times 3 \times 7\]
\[\therefore LCM = 2520\]

Therefore, the LCM of \[40\], \[42\] and \[45\] is \[2520\].

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 and easiest way to solve the problem.