MHB -c6. Find the GCF and the LCF of A and B.

[karush]

Find the greatest common factor and the least common multiple of A and B.
Write your answers as a product of powers of primes in increasing order.
$A=2^3 3 \cdot 5^3 \cdot 11^2 \cdot 13 $
$B = 2 \cdot 3^3 \cdot 7^3 \cdot 11^2 \cdot 13^3 $
$GCF(A,B) = \boxed{?}$
$LCM(A,B) = \boxed{?}$

ok apparently this is just by observation
but its kinda subtle so I went with $GCF(A,B) = \boxed{11^2}$ $LCM(A,B) = \boxed{2}$
hopefully

 

[HOI]

I am afraid you are misunderstanding "greatest common factor". That is not a single prime that is in both numbers, it is the largest number that evenly divides both. I see that A has [math]2^3[/math] as a factor and B has 2 as a factor so the GCF has 2 as a factor. A has 3 as a factor and B has [math]3^3[/math] so the GCF has 3 as a factor. A has [math]5^3[/math] as a factor but B does not have a power of 5 as a factor so the GCF does not have a power of 5 as a factor. B has [math]7^3[/math] as a factor but A does not have a power of 7 as a factor so the GCF does not have a power of 7 as a factor. Both A and B have [math]11^2[/math] as a factor so the GCF has [math]11^2[/math] as a factor. A has 13 as a factor and B has [math]13^3[/math] as a factor so the GCF has13 as a factor.

The GCF is [math](2)(3)(11^2)(13)[/math]

For the GCF we took the lowest power of each prime. For the LCM (least common multiple) take the highest power. The LCM of A and B is [math](2^3)(3^3)(5^3)(7^3)(11^2)(13^3)[/math].