# How are GCF and LCM alike?

Aug 24, 2016

They are not really alike at all...

#### Explanation:

GCF and LCM are actually not alike at all and they are often easily confused.

We tend to hear GREATEST and LOWEST when we hear these names.

GREATEST makes us think BIG, while LOWEST makes us think small.

We should actually be more interested in the words FACTOR and MULTIPLE.

FACTORS are small numbers which divide INTO a bigger number.
MULTIPLES are big numbers which are divisible BY factors.

MULTIPLES of $24 \rightarrow 24 , 48 , 72 , 96 , 120 , 144 , 168 , 192 \ldots \ldots$

FACTORS of $24 \rightarrow 1 , 2 , 3 , 4 , 6 , 8 , 12 , 24$

Multiples continue to infinity, there is no such thing as the Greatest Multiple.

Factors are finite. 24 has exactly 8 factors. The lowest common factor will always be 1.

The Greatest Common FACTOR of several numbers is the highest FACTOR (ie a small value) which divides INTO all of the given numbers. The largest possible value is the smallest of the given numbers.

The GCF of $15 , 30 , 75 \mathmr{and} 105 \text{ is } 15$
(15 is small)

The Lowest Common MULTIPLE of several numbers is the first MULTIPLE (a big number) which is divisible BY all of the given numbers.

The LCM of $15 , 30 , 75 \mathmr{and} 105 \text{ is } 1050$
(1050 is big)

Prime factors is a good way of finding the GCF and LCM

$\textcolor{w h i t e}{\times \times \times} 15 = \textcolor{w h i t e}{\times x} 3 \times 5$
$\textcolor{w h i t e}{\times \times \times} 30 = 2 \times 3 \times 5$
$\textcolor{w h i t e}{\times \times \times} 75 = \textcolor{w h i t e}{\times x} 3 \times 5 \times 5$
$\textcolor{w h i t e}{\times \times . .} 105 = \textcolor{w h i t e}{\times x} 3 \times 5 \textcolor{w h i t e}{\times} \times 7$

$G C F = \textcolor{w h i t e}{\times \times \times x} 3 \times 5 \textcolor{w h i t e}{\times \times \times \times} = 15$

$L C M = \textcolor{w h i t e}{\times \times} 2 \times 3 \times 5 \times 5 \times 7 \textcolor{w h i t e}{\times} = 1050$