Please help.....L.C.M problem.?

Question

How to find the L.C.M of a + b + c............................................
............................................ ___ ___ ___ ?
............................................ a+b b+c c+a

It is,a upon a+b,plus b upon b+c,plus c upon c+a

Please explain in detail with proper steps and instructions.

Answer 1

The LCM of these three denominators would be the product of the three. The LCM in general would be the common number that you would have if you counted by your denominators. For example, if your denominators were 2, 4, and 6, you'd have 12 as your LCM:

Counting by 2's - 2, 4, 6, 8, 10, 12, 14...
Counting by 4's - 4, 8, 12, 16, 20...
Counting by 6's - 6, 12, 18, 24...

The first number in common with all three is 12, so that is your LCM.

With variables, you can't find the LCM this way, so you'll just have to multiply them together to find the LCM. So,

(a + b)(b + c)(c + a) = (ab + ac + b^2 + bc)(c + a)
= abc + ac^2 + b^2c + bc^2 + a^2b + a^2c + ab^2 + abc
= a^2b + a^2c + ab^2 + b^2c + ac^2 + bc^2 + 2abc

Now if you're looking to combine these fractions, you'll have to multiply the numerators by the factors you multiplied by. For example, a/(a + b) was multiplied by (b + c)(c + a), so you'll have to multiply a by that in order to get your new numerator:

a(b + c)(c + a) = (ab + ac)(c + a)
= abc + a^2b + ac^2 + a^2c

You'd have to do that for the other two factors as well. Once you do that, you'd have to combine all of your like terms in order to come up with your answer.