MHB Calculating LCD and GCD of 2 Sets of Numbers

[karush]

Determine
$\textit{gcd}(2^4 \cdot 3^2 \cdot 5 \cdot 7^2,\quad 2 \cdot 3^3 \cdot 7 \cdot 11)$
and
$\textit{lcm}(2^3 \cdot 3^2 \cdot 5,\quad 2 \cdot 3^3 \cdot 7 \cdot 11)$

ok the example appeared to have combine the 2 sets on gcd but I am still ?

there is no book answer for this

 

[skeeter]

Greatest Common divisor ... think about how you would factor out the common terms from both prime decompositions if they were added

$(2^4 \cdot 3^2 \cdot 5 \cdot 7^2) + (2 \cdot 3^3 \cdot 7 \cdot 11)$

${\color{red}(2 \cdot 3^2 \cdot 7)} \bigg[(2^3 \cdot 5 \cdot 7)+ ( 3 \cdot 11) \bigg]$Least Common Multiple ... think about obtaining a common denominator if both prime factor decompositions were denominators of two fractions

$\dfrac{x}{2^3 \cdot 3^2 \cdot 5} + \dfrac{y}{2 \cdot 3^3 \cdot 7 \cdot 11}$

$\dfrac{x(3 \cdot 7 \cdot 11) + y(2^2 \cdot 5)}{\color{red} 2^3 \cdot 3^3 \cdot 5 \cdot 7 \cdot 11}$

 

[karush]

well that makes a lot more sense

i don't think there is any need to multiple these out