Find the GCF and/or the LCM?

Question

FIND THE GCF AND LCM FOR THE FOLLOWING
1) -14,28
2) 36,45
3) -28,32

PLEASE ALSO EXPLAIN HOW TO FIND THE GCF AND LCM FOR SOMETHING THAT HAS A NEGATIVE. I.E PROBLEMS 1 &3

Answer 1

Ok, for negatives, you need to remember that the gcf is the largest number that can divide all of the numbers, so it would have to be positive, since, positive is larger than negative. For lcm, you want the smallest number that all the numbers can go into, and, since negative is smaller than positive, the lcm would be negative.

To find gcf and lcm, you begin by factoring all the numbers into their prime factors. But here's where it's different. For the gcf, you pick up the lowest powers of only the factors that occur in all numbers. That is, the intersection of the sets of factors. For the lcm, you pick up the highest powers of all the factors that occur in any of the numbers. That is, the union of the sets of factors. To understand why this is, think about the definitions of gcf and lcm. So here we go ...

-14, 28
-14 = -(2 x 7) 28 = 2 x 2 x 7
lcm is the smallest number which both of them go into, so that would be -(2x2x7) = -28
gcf would be the greatest number that divides both of them, so that would be the only factor they have in common, 2.

36. 45
36 = 2x2x3x3 45 = 3x3x5
lcm is the smallest number that they both go into, so you need both 2's, both 3's, and the 5. 2x2x3x3x5 = 180
gcf is the biggest number that goes into both, they have the 3x3 in common, so the gcf would be 3x3 = 9

-28, 32
-28 = -(2x2x7) 32 = 2x2x2x2x2
For the smallest number that they both go into, collect all 5 of the 2's, and the 7. 2x2x2x2x2x7= 224 (I think, I don't have a calculator handy and I did it in my head) That's the lcm, only it needs to be negative because negative is less than positive so it would be -224.
For the biggest number that goes into both, they have 2x2 = 4 in common so the greatest common factor is 4

Answer 2

the gcf of #2 is 9
the lcm is 180