# Question fb025

Oct 12, 2017

see below

#### Explanation:

The lowest common multiple of two numbers is the smallest number both will divide into exactly.

For instance

$\lcm \left(2 , 3\right) = 6$

because 6 is the smallest number that BOTH $2 \text{ & } 3$ will divide into

there are several methods to go about find the $L C M$
here I will illustrate just two

method 1

The simplest way is just to write down the multiples of both numbers and then pick out the common ones and hence the lowest is identified easily.

lcm(4,12)

multiples of $4 : \left\{4 , 8 , \textcolor{red}{12} , 16 , 20 , \textcolor{red}{24} , 28 , 32 , \textcolor{red}{36} \ldots\right\}$

multiples of $12 : \left\{\textcolor{red}{12 , 24 , 36} , \ldots\right\}$

the common multiples are highlighted in red

common multiples$\left\{12 , 24 , 36 , \ldots\right\}$

$\therefore \lcm \left(4 , 12\right) = 12.$

method 2

for two numbers $a , b$

The second method uses the relationship

$a \times b = h c f \left(a , b\right) \times \lcm \left(a , b\right)$

this is particularity useful when the numbers are too large for listing the multiples.
so

$\lcm \left(25 , 35\right)$

$h c f \left(25 , 35\right) = 5$

$\therefore 25 \times {\cancel{35}}^{7} = \cancel{5} \times \lcm \left(25 , 35\right)$

$\lcm \left(25 , 35\right) = 25 \times 7 = 175$

the third method using prime factors has been covered elsewhere

Oct 12, 2017

L C M : The smallest positive number that is a multiple of two are more numbers.

To find the L C M of 4, 10 :
Multiples of 4 are 4 8 12 16 20 24 28 32 36 40 and so on.
Factors of 10 = 10 20 30 40 and so on.
There is a match @ 20 which is the L C M

#### Explanation:

L C M : The smallest positive number that is a multiple of two are more numbers.
L C M of 3 & 5 is 15, because 15 is a multiple of 3 & 5. Other common multiples include 30, 45, etc., but they are not the smallest.

To find the L C M of 4, 10 :
Multiples of 4 are 4 8 12 16 20 24 28 32 36 40 and so on.
Factors of 10 = 10 20 30 40 50 60 and so on.
There is a match @ 20 which is the least common multiple.