Common Division Method

A method of dividing the numbers commonly by a number to find the least common multiple (LCM) of them is called the common division method.

Procedure

The common division method is a simple method to find the lowest common multiple of the given numbers. In this method, we divide the given numbers commonly by a suitable number. There are four steps involved in this method.

1. Draw a perpendicularly intersected lines.
2. Write all the numbers by a comma separation at right side of the vertical line and over the horizontal line. Take a suitable number, which can divide at least two of the given numbers and write it at left side of the vertical line and over the horizontal line. Write the quotients and undivided numbers under the horizontal line at the corresponding positions.
3. If at least two numbers from the quotients and undivided numbers can be divisible by a number, then draw a horizontal under them and extent the vertical line to split it as two parts.
4. Repeat the same procedure until we get the numbers in which at least two numbers are not divisible by any number. Finally, calculate the product of the divisors and the undivided numbers for getting the LCM of the given numbers.

Example

Find the L.C.M of $4, 8, 12, 16, 24$ and $36$

Firstly, draw a perpendicularly intersected lines.

$\,\,\,$	$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
	$\,$

Write the given numbers $4,$ $8,$ $12,$ $16,$ $24$ and $36$ at right side of the vertical line but above the horizontal line.

$\,\,\,$	$4, 8, 12, 16, 24, 36$
	$\,$

All the given numbers are divisible by $2$. So, write it at left side of the vertical line and above the horizontal line.

$2$	$4, 8, 12, 16, 24, 36$
	$\,$

Every number is commonly divisible by $2$ and write each quotient at the corresponding position under the horizontal line.

$2$	$4, 8, 12, 16, 24, 36$
	$2, 4, \,\,\, 6, \,\,\, 8, 12, 18$

The numbers $2, 4, 6, 8, 12$ and $18$ can also be divisible by $2$. So, draw a horizontal line again and extend the vertical line to split it.

$2$	$4, 8, 12, 16, 24, 36$
	$2, 4, \,\,\, 6, \,\,\, 8, 12, 18$
$\,$	$\,$

Now, write $2$ at left side of the vertical line and over the horizontal line, and then write the quotients below the horizontal line.

$2$	$4, 8, 12, 16, 24, 36$
$2$	$2, 4, \,\,\, 6, \,\,\, 8, 12, 18$
$\,$	$1, 2, \,\,\, 3, \,\,\, 4, \,\,\, 6, \,\,\, 9$

In the numbers $1, 2, 3, 4, 6$ and $9,$ the numbers $2, 4$ and $6$ are divisible by $2$ and the remaining numbers are indivisible. So, repeat the same procedure. Write the quotients in the respective positions but write the undivided numbers as they are, below the horizontal line.

$2$	$4, 8, 12, 16, 24, 36$
$2$	$2, 4, \,\,\, 6, \,\,\, 8, 12, 18$
$2$	$1, 2, \,\,\, 3, \,\,\, 4, \,\,\, 6, \,\,\, 9$
$\,$	$1, 1, \,\,\, 3, \,\,\, 2, \,\,\, 3, \,\,\, 9$

The numbers $3, 3$ and $9$ are divisible by $3$ and the remaining numbers are indivisible. So, repeat the process one more time.

$2$	$4, 8, 12, 16, 24, 36$
$2$	$2, 4, \,\,\, 6, \,\,\, 8, 12, 18$
$2$	$1, 2, \,\,\, 3, \,\,\, 4, \,\,\, 6, \,\,\, 9$
$3$	$1, 1, \,\,\, 3, \,\,\, 2, \,\,\, 3, \,\,\, 9$
$\,$	$1, 1, \,\,\, 1, \,\,\, 2, \,\,\, 1, \,\,\, 3$

There are no at least two divisible numbers in the numbers $1, 1, 1, 2, 1$ and $3$. So, we can stop the procedure.

In this example, the divisors are $2, 2, 2$ and $3$ and the undivided numbers $1, 1, 1, 2, 1$ and $3$. The product of them is called least common multiple.

$L.C.M$ $\,=\,$ $2$ $\times$ $2$ $\times$ $2$ $\times$ $3$ $\times$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $2$ $\times$ $1$ $\times$ $3$

$\therefore \,\,\,\,\,\,$ $L.C.M \,=\, 144$

In this way, the LCM of the given numbers is calculated in the common division method.

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