Question
Two numbers have prime factorizations of 2 2 · 3 · 5 and 2 · 3 2 · 5.
Which expressions will give the greatest common factor and the least common multiple?
GCF 2 · 3 · 5; LCM 2 2 · 3 2 · 5
GCF 2 2 · 3 2; LCM 2 2 · 3 2 · 5 2
GCF 2 · 3; LCM 2 · 3 · 5
GCF 2 2 · 3 2 · 5; LCM 2 · 3
Which expressions will give the greatest common factor and the least common multiple?
GCF 2 · 3 · 5; LCM 2 2 · 3 2 · 5
GCF 2 2 · 3 2; LCM 2 2 · 3 2 · 5 2
GCF 2 · 3; LCM 2 · 3 · 5
GCF 2 2 · 3 2 · 5; LCM 2 · 3
Answers
GPT-4o mini
To find the GCF (greatest common factor) and LCM (least common multiple) of the two numbers with the given prime factorizations, let's first write down the prime factorizations clearly:
1. The first number's factorization is: \( 2^2 \cdot 3^1 \cdot 5^1 \)
2. The second number's factorization is: \( 2^1 \cdot 3^2 \cdot 5^1 \)
### Finding the GCF (Greatest Common Factor):
To find the GCF, we take the lowest power of each prime factor present in both factorizations.
- For the prime \(2\): the lowest power is \(2^1\).
- For the prime \(3\): the lowest power is \(3^1\).
- For the prime \(5\): the lowest power is \(5^1\).
So, the GCF is:
\[
GCF = 2^1 \cdot 3^1 \cdot 5^1 = 2 \cdot 3 \cdot 5
\]
### Finding the LCM (Least Common Multiple):
To find the LCM, we take the highest power of each prime factor present in the factorizations.
- For the prime \(2\): the highest power is \(2^2\).
- For the prime \(3\): the highest power is \(3^2\).
- For the prime \(5\): the highest power is \(5^1\).
So, the LCM is:
\[
LCM = 2^2 \cdot 3^2 \cdot 5^1
\]
### Final Results:
- GCF = \( 2^1 \cdot 3^1 \cdot 5^1 \) = \( 2 \cdot 3 \cdot 5 \)
- LCM = \( 2^2 \cdot 3^2 \cdot 5^1 \)
Now we can match these results to the provided options:
1. **GCF 2 · 3 · 5**; **LCM 2² · 3² · 5**
2. **GCF 2² · 3²**; **LCM 2² · 3² · 5²**
3. **GCF 2 · 3**; **LCM 2 · 3 · 5**
4. **GCF 2² · 3² · 5**; **LCM 2 · 3**
From our calculations, the correct expressions are:
**GCF 2 · 3 · 5** (which corresponds to 2, in expanded prime factorization as \(2^1 \cdot 3^1 \cdot 5^1\)) and **LCM 2² · 3² · 5** (which corresponds to \( 2^2 \cdot 3^2 \cdot 5^1\)).
Thus, the correct choice is:
**GCF 2 · 3 · 5; LCM 2² · 3² · 5**.
1. The first number's factorization is: \( 2^2 \cdot 3^1 \cdot 5^1 \)
2. The second number's factorization is: \( 2^1 \cdot 3^2 \cdot 5^1 \)
### Finding the GCF (Greatest Common Factor):
To find the GCF, we take the lowest power of each prime factor present in both factorizations.
- For the prime \(2\): the lowest power is \(2^1\).
- For the prime \(3\): the lowest power is \(3^1\).
- For the prime \(5\): the lowest power is \(5^1\).
So, the GCF is:
\[
GCF = 2^1 \cdot 3^1 \cdot 5^1 = 2 \cdot 3 \cdot 5
\]
### Finding the LCM (Least Common Multiple):
To find the LCM, we take the highest power of each prime factor present in the factorizations.
- For the prime \(2\): the highest power is \(2^2\).
- For the prime \(3\): the highest power is \(3^2\).
- For the prime \(5\): the highest power is \(5^1\).
So, the LCM is:
\[
LCM = 2^2 \cdot 3^2 \cdot 5^1
\]
### Final Results:
- GCF = \( 2^1 \cdot 3^1 \cdot 5^1 \) = \( 2 \cdot 3 \cdot 5 \)
- LCM = \( 2^2 \cdot 3^2 \cdot 5^1 \)
Now we can match these results to the provided options:
1. **GCF 2 · 3 · 5**; **LCM 2² · 3² · 5**
2. **GCF 2² · 3²**; **LCM 2² · 3² · 5²**
3. **GCF 2 · 3**; **LCM 2 · 3 · 5**
4. **GCF 2² · 3² · 5**; **LCM 2 · 3**
From our calculations, the correct expressions are:
**GCF 2 · 3 · 5** (which corresponds to 2, in expanded prime factorization as \(2^1 \cdot 3^1 \cdot 5^1\)) and **LCM 2² · 3² · 5** (which corresponds to \( 2^2 \cdot 3^2 \cdot 5^1\)).
Thus, the correct choice is:
**GCF 2 · 3 · 5; LCM 2² · 3² · 5**.