find least common multiple of 6, 7, 8
prime factorization: (2^(3))(3)(7) = 168
I don't quite get how 3 was picked from 6.
I think 2^3 is from 8 and 7 is from 7 but 6 is 3 x 2 = 6
But why not choose 2 instead of 3?
6 = 2x3
7 = 7
8 = 2x2x2
for the least common multiple each factor above must show up, so you need
2x2x2x3x7
(the 2 of the 2x3 was already taken care of in the 2x2x2, so we don't have to repeat it)
2x2x2x3x7 = 168
To find the prime factorization of a number, we decompose it into its prime factors. In the case of 6, we notice that it can be expressed as the product of 2 and 3: 6 = 2 * 3.
The reason we choose 3 instead of 2 for the prime factorization is that we want to find the LCM (Least Common Multiple) of 6, 7, and 8. To calculate the LCM, we need to find the highest exponent for each prime factor that appears in any of the three numbers.
For example, let's look at the number 8. Its prime factorization is: 8 = 2^3. The exponent 3 represents the highest power of 2 that divides evenly into 8.
Now, when it comes to the number 6, we can express it as the product of 2 and 3. However, the highest power of 2 that divides evenly into 6 is 2^1, not 2^3 as in the case of 8. Therefore, we only include 2^1 in the prime factorization of 6.
Finally, the prime factorization of 7 is simply 7, as it is already a prime number.
After determining the prime factorization of each number, we multiply the highest exponent of each prime factor to find the LCM. In this case, it is (2^3) * (3^1) * 7 = 168.
So, the main reason we choose 3 instead of 2 for the prime factorization of 6 is that we are looking for the highest power of each prime factor that appears in any of the given numbers in order to find their LCM.