# Question #b75c8

Oct 19, 2017

Both depend on factors of the numbers.

#### Explanation:

The GCF is the greatest common factor. To find the GCF factor all the numbers and look for the largest factor than is the same for all the numbers.

The LCM is the smallest common multiple of all the numbers. To find the LCM factor all the numbers. Then multiply all the unique factors. If there is a 2 in four of the numbers the 2 only needs to be used once in finding the Least Common Multiple not four times.

Solving for both the GCF and LCM depend on finding the factors of all the numbers involved. This is one way the GCF and LCM are similar.

Oct 19, 2017

G C F is the greatest common factor or greatest common divisor which will contain only the common factors that are found between 2 or more terms.

L C M is the least common multiple and is the least number which is divisible by 2 or more terms.

#### Explanation:

G C F is the greatest common factor or greatest common divisor which will contain only the common factors that found between 2 or more terms.

L C M is the least common multiple and is the least number which is divisible by 2 or more terms.

Eg:
Taking two numbers 12, 18

Factors for $12 = 2 \cdot 2 \cdot 3$
Factors for $18 = 2 \cdot 3 \cdot 3$

G C F :
Factors for 12 = $\textcolor{red}{2} \cdot 2 \cdot \textcolor{red}{3}$
Factors for 18 = $\textcolor{red}{2 \cdot 3} \cdot 3$

$G C F = 2 \cdot 3 = 6$ This is the great common factor for 12, 18.

L C M :
Factors for 12 = $\textcolor{red}{2} \cdot 2 \cdot \textcolor{red}{3}$
Factors for 18 = $\textcolor{red}{2 \cdot 3} \cdot 3$

L C M = $2 \cdot 2 \cdot 3 \cdot 3 = 36$. This is the least multiple of both 12 & 18.

Dec 29, 2017

It is easy to define, when expressing numbers by prime factorization with powers.

#### Explanation:

Let's first see divisibility relation in this way:

$144 = {2}^{4} \cdot {3}^{2}$
$12 = {2}^{2} \cdot {3}^{1}$

144 is divisible by 12 because all exponents in factorization of 144 are greater than or equal to corresponding exponents of 12.

GCD and LCM are mostly applied to pairs of numbers, but work as well for 3 or more.

$x = \gcd \left(84 , 120 , 144\right)$
This means that each of these 3 numbers is divisible by $x$. Let's look at the factorization:
$84 = {2}^{2} \cdot {3}^{1} \cdot {5}^{0} \cdot {7}^{1}$
$120 = {2}^{3} \cdot {3}^{1} \cdot {5}^{1} \cdot {7}^{0}$
$144 = {2}^{4} \cdot {3}^{2} \cdot {5}^{0} \cdot {7}^{0}$
Exponents of $x$ factorization must be less than or equal to exponents of 84, 120 and 144. At the same time $x$ must be greatest possible, so:
$x = {2}^{\min \left(2 , 3 , 4\right)} \cdot {3}^{\min \left(1 , 1 , 2\right)} \cdot {5}^{\min \left(0 , 1 , 0\right)} \cdot {7}^{\min \left(1 , 0 , 0\right)}$
$x = {2}^{2} \cdot {3}^{1} \cdot {5}^{0} \cdot {7}^{0} = 12$

$y = \lcm \left(84 , 120 , 144\right)$
This means that $y$ is divisible by each of these 3 numbers. This time exponents of $y$ factorization must be greater than or equal to exponents of 84, 120 and 144. At the same time $y$ must be least possible, so:
$y = {2}^{\max \left(2 , 3 , 4\right)} \cdot {3}^{\max \left(1 , 1 , 2\right)} \cdot {5}^{\max \left(0 , 1 , 0\right)} \cdot {7}^{\max \left(1 , 0 , 0\right)}$
$y = {2}^{4} \cdot {3}^{2} \cdot {5}^{1} \cdot {7}^{1} = 8 \cdot 9 \cdot 10 \cdot 7 = 72 \cdot 70 = 5040$

The exponents 0 and 1 can be omitted while writing for comfort, but must be considered, when comparing numbers (up to largest prime with nonzero exponent).

Summing up, GCD is found by taking $\min$ of the exponents and LCM is found by taking $\max$ of the exponents.