Apply distributive property to factor out the greatest common factor for the numbers 30 and 48. show your work please.

1 answer

To factor out the greatest common factor (GCF) of the numbers 30 and 48 using the distributive property, we first need to determine what the GCF is.

Step 1: Find the GCF of 30 and 48.

Step 1.1: Find the prime factorization of each number.

  • For 30:

    • \(30 = 2 \times 3 \times 5\)
  • For 48:

    • \(48 = 2^4 \times 3\)

Step 1.2: Identify the common factors.

  • The common prime factors between 30 and 48 are \(2\) and \(3\).

Step 1.3: Determine the lowest power of each common factor.

  • For \(2\): The minimum power is \(2^1\) (from 30).
  • For \(3\): The minimum power is \(3^1\) (common in both).

Step 1.4: Calculate the GCF. The GCF is calculated as: \[ GCF = 2^1 \times 3^1 = 2 \times 3 = 6 \]

Step 2: Factor out the GCF from both numbers.

Now that we know the GCF is 6, we can express each of the original numbers (30 and 48) as a product involving this GCF:

  1. For 30: \[ 30 = 6 \times 5 \]
  2. For 48: \[ 48 = 6 \times 8 \]

Step 3: Use the distributive property.

We can now combine the factors using the distributive property: \[ 30 + 48 = (6 \times 5) + (6 \times 8) \]

Using the distributive property, we factor out the GCF: \[ 30 + 48 = 6(5 + 8) \]

Final Result:

Thus, by applying the distributive property to factor out the greatest common factor, we have: \[ 30 + 48 = 6(5 + 8) = 6 \times 13 \] This shows how you can factor out the GCF of 30 and 48 using the distributive property.