To factor out the greatest common factor (GCF) using the distributive property for the numbers 30 and 48, we first need to determine the GCF of these two numbers.
Step 1: Find the GCF of 30 and 48.
We start by finding the prime factorization of each number.
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The prime factorization of 30 is: \[ 30 = 2 \times 3 \times 5 \]
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The prime factorization of 48 is: \[ 48 = 2^4 \times 3 \]
Step 2: Identify common factors.
Now we identify the common prime factors:
- Both 30 and 48 include the factors \(2\) and \(3\).
Step 3: Determine the smallest exponent for each common factor.
- For \(2\): The smallest power in both factorizations is \(2^1\).
- For \(3\): The smallest power in both factorizations is \(3^1\).
Step 4: Calculate the GCF.
Now we multiply these common factors together to find the GCF:
\[ \text{GCF} = 2^1 \times 3^1 = 2 \times 3 = 6 \]
Step 5: Factor out the GCF using the distributive property.
Now we can express both numbers (30 and 48) as their GCF multiplied by another factor:
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For 30: \[ 30 = 6 \times 5 \]
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For 48: \[ 48 = 6 \times 8 \]
Final Expression:
Thus, we can use the distributive property to express the relationship between these two numbers as follows:
\[ 30 + 48 = 6 \times 5 + 6 \times 8 \]
Using the distributive property, we can factor out the GCF:
\[ 30 + 48 = 6 \times (5 + 8) \]
Calculating the expression inside the parentheses:
\[ 5 + 8 = 13 \]
So, we have:
\[ 30 + 48 = 6 \times 13 \]
In summary, the greatest common factor of 30 and 48 is 6, and we can factor the sum \(30 + 48\) as:
\[ 30 + 48 = 6 \times 13 \]