I am stumped by these two LCM questions for some reason.

Find The LCM for
#1
2(R-7) and 14(R-7)

#2
(1+4r), (1-16r^2), and (1-4r)

It seems like #1 is already at the LCM
and #2 would be (1+4r)(1-4r)
Am I even vaguely correct?
Thanks for your help.

As you probably know, if one number is a multiple of the other, the bigger number is the LCM of the two.

Post your exact answer to #1 to confirm if you wish.

For #2, you are more than vaguely correct, but you do not appear to have any justification of your response.

You can confirm your thoughts by dividing the LCM by each of the three values. If they all divide without remainder, and there is no common factor between the quotients, the answer is correct.

For example,
If 60 is the LCM of 5,6,15, then
60/5=12 has no remainder
60/6=10 has no remainder
60/15=5 has no remainder.
However,
5,10 and 12 have a common factor of 2, so 60 is not the LCM, but 60/2=30 is!

Feel free to post your justification of your answer along the lines above.

To find the Least Common Multiple (LCM) for the given expressions, you need to factorize each expression and identify the highest powers of all the factors present. Let's break down each question step by step:

#1:
Given expressions: 2(R-7) and 14(R-7)

To find the LCM, we can first simplify the expressions:
2(R-7) = 2R - 14
14(R-7) = 14R - 98

Now let's factorize each expression:
2R - 14 = 2(R - 7)
14R - 98 = 14(R - 7)

As you correctly noted, the expressions are already in the same form (R - 7). Therefore, the LCM in this case is simply (R - 7).

#2:
Given expressions: (1 + 4r), (1 - 16r^2), and (1 - 4r)

To find the LCM, first, let's check if any factor is repeated across multiple expressions. In this case, the factor (1) appears in all three expressions. Next, let's factorize the expressions:
1 + 4r = 1 + 4r
1 - 16r^2 = (1 + 4r)(1 - 4r)
1 - 4r = 1 - 4r

When you factorized (1 - 16r^2) as (1 + 4r)(1 - 4r), you made a mistake by using the difference of squares formula. The correct factorization should be (1 - 16r^2) = (1 + 4r)(1 - 4r^2).

Now, let's list all the factors present in these expressions:
(1 + 4r), (1 - 16r^2), (1 - 4r^2)

To find the LCM, you need to identify the highest power of each factor. In this case, the factors are (1 + 4r), (1 - 4r^2), and (1 - 16r^2).

The LCM will be the product of the highest powers for each factor. Therefore, the LCM for this question is: (1 + 4r)(1 - 4r^2)(1 - 16r^2).

So, to summarize:
#1: The LCM is (R - 7).
#2: The LCM is (1 + 4r)(1 - 4r^2)(1 - 16r^2).

Remember to carefully factorize the expressions and consider the highest powers of each factor when finding the LCM.