Question

How do I write the sum of two numbers as the product of their GCF and another sum

Answer 1

Let's assume we were given 2 numbers: 15 and 30. Their sum is:
$15+30 = 45$
We want to express it as the product of GCF and another sum.
15 is divisible by: 1, 3, 5, 15
30 is divisible by: 1, 2, 3, 5, 6, 10, 15, 30
The greatest number that appears in 2 series is 15.
$\frac{15}{15} = 1$
$\frac{30}{15} = 2$
In this case sum of two numbers can always be written as:
$15+30 = (15\cdot 1) + (15\cdot 2) = 15\cdot (1+2) = 15\cdot 3 = 45$

Answer 2

1) Start by calculating the greatest common factor (GCF)

For example, take the sum 130 + 26.

Calculate the GFC:

i) prime decompostion:

130 = 2×5×13

26 = 2×13

ii) the GCF is the product of the common factors each raised to the lowest power: 2×13 = 26.

2) Use the GCF as factor of the two numbers: write the common factor, open parenthesis, and write inside the parenthesis the quotient of each number when divided by the GCF.

Following our example with the numbers 130 and 26:

26 ( 130÷26 + 26÷26) = 26 (5 + 1)

The result is 26 (5 + 1).

3) This calculation is named factoring and is the operation opposite to the use of the distributive property.

4) You can verify that 130 + 26 = 26 ( 5 + 1)

130 + 26 = 156

26 (5 + 1) = 26 × 6 = 156