Let me know if this is correct :

(1)Finding the lcm using which ever method for:5,15 , and 20

8:2x2x2x3x5x5
15:2x2x2x3x5x5
20:2x2x2x3x5x5
---------------
LCM:2x2x2x3x5x5
After simplification LCM:120

(2)Find the GCF for each of the following numbers: 36, 64, 180

Prime Factors
36=2x2x3x3x1
64=2x2x2x2x2x2x1
180=2x2x3x3x5x1
Common Prime Factors
36= 2x2x2
64=2x2x2
180=2x2x2

GCF=4

I am not certain what your logic is, but the answers are correct.

The logic used in the given solutions is correct. Let me explain how to find the LCM and GCF of the given numbers:

(1) Finding the LCM:
To find the LCM (Least Common Multiple) of the numbers 5, 15, and 20, you can use the prime factorization method.

First, factorize each number into its prime factors:
5 = 5 (since 5 is a prime number)
15 = 3 × 5
20 = 2 × 2 × 5

Next, you take the highest power of each prime factor that appears in any of the numbers. In this case, the prime factors are 2, 3, and 5.

2 appears twice in the factorization of 20, so we take 2^2.
3 appears once in the factorization of 15, so we take 3^1.
5 appears once in the factorization of 15 and 20, so we take 5^1.

Multiplying these together, we get:
LCM = 2^2 × 3^1 × 5^1 = 4 × 3 × 5 = 60.

So, the correct LCM of 5, 15, and 20 is 60. Your calculation of 120 is incorrect.

(2) Finding the GCF:
To find the GCF (Greatest Common Factor) of the numbers 36, 64, and 180, you can again use the prime factorization method.

Factorize each number into its prime factors:
36 = 2 × 2 × 3 × 3
64 = 2 × 2 × 2 × 2 × 2 × 2
180 = 2 × 2 × 3 × 3 × 5

Now, you look for the common prime factors among these numbers.

The common prime factors are:
2 appears twice in the factorization of 36 and 64, but only once in the factorization of 180.
So, we take 2 × 2 = 4.

The GCF is 4.

Therefore, your calculation of GCF=4 is correct.