Arithmetic/Lowest Common Multiple

To find the Lowest Common Multiple (LCM) of several numbers, we first express each number as a product of its prime factors.

For example, if we wish to find the LCM of 60, 12 and 102 we write

${\begin{matrix}60=2^{2}\cdot 3\cdot 5\\12=2^{2}\cdot 3\\102=2\cdot 3\cdot 17\end{matrix}}$

The product of the highest power of each different factor appearing is the LCM.

For example in this case, $2^{2}\cdot 3\cdot 5\cdot 17=1020$. You can see that 1020 is a multiple of 12, 60 and 102 ... the lowest common multiple of all three numbers.

Another example: What is the LCM of 36, 45, and 27?

Solution: Factorise each of the numbers

${\begin{matrix}36=2^{2}\cdot 3^{2}\\45=5\cdot 3^{2}\\27=3^{3}\end{matrix}}$

The product of the highest power of each different factor appearing is the LCM, i.e.;

$2^{2}\cdot 5\cdot 3^{3}=540$

Properties of the LCM

If the LCM of the numbers is found and 1 is subtracted from the LCM then the remainder when divided by each of the numbers whose LCM is found would have a remainder that is 1 less than the divisor. For example if the LCM of 2 numbers 10 and 9 is 90. Then 90-1=89 and 89 divided by 10 leaves a remainder of 9 and the same number divided by 9 leaves a remainder of 8.