Prealgebra for Two-Year Colleges/Appendix (procedures)/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, $$2^2\cdot3\cdot5\cdot17=1020$$. 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

$$ \begin{matrix} 36=2^2\cdot3^2\\ 45=5\cdot3^2 \\ 27=3^3 \end{matrix} $$

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

$$2^2\cdot5\cdot3^3=540$$

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.