Question 1: What is the greatest common factor (GCF) of the given numbers 40 and 75?
To find the GCF, we need to find the prime factorization of each number:
- 40 = \(2^3 \times 5^1\)
- 75 = \(3^1 \times 5^2\)
The common prime factor is 5. Therefore, the GCF is:
\[ GCF(40, 75) = 5 \]
Since none of the provided options (22, 33, 55, 77) correspond to the actual GCF, it seems there might be an error in the provided responses.
Question 2: What is the least common multiple (LCM) of the numbers 6 and 8?
To find the LCM, we start with the prime factorization of each number:
- 6 = \(2^1 \times 3^1\)
- 8 = \(2^3\)
Next, we take the highest power of each prime factor:
- For \(2\): The highest power is \(2^3\)
- For \(3\): The highest power is \(3^1\)
Now, we multiply these together:
\[ LCM(6, 8) = 2^3 \times 3^1 = 8 \times 3 = 24 \]
Again, since none of the provided options (1818, 2424, 2727, 3232) are correct, there may be an error in the responses given. The LCM of 6 and 8 is 24.
In summary:
- GCF of 40 and 75 is 5
- LCM of 6 and 8 is 24