We can combine a few concepts together to start getting towards a new property of arithmetic...
Question
What are the factors of 8 and 6? What factors is common to both? Can we express 8 + 6 = 14 using a multiplication statement?
What are multiples of 3?Answer
Factors of 8 are 1, 2, 4, and 8. Factors of 6 are 1, 2, 3, 6. The number 2 is common to them both. We can express 8 + 6 = 14 as 2 x (4 + 3) = 2 x 7 = 14.
A few multiples of 3 are 3, 6, 9, 12, and 15.Analysis
Let's first talk about what a Factor is.
A factor is a number, or term, that when multiplied by another term (and it can be the same one), the product is the number we're getting factors for.
Let's look at 6. What numbers are factors of 6? (Or another way to ask this is What are the multiplications that get us to 6). If we keep the factors to whole numbers, we'll get:
1 x 6 = 6
2 x 3 = 6
So 1, 2, 3, and 6 are factors of 6.
We can do the same thing for the number 8:
1 x 8 = 8
2 x 4 = 8
So 1, 2, 4, and 8 are factors of 8.
Now we have those, which factors, if any, are common to them both? The number 2.
Let's now change focus and look at the addition problem:
8 + 6 = 14
Let's rewrite the 8 and 6 using the factors (and in particular the common factor 2):
(2 x 4) + (2 x 3) = 14
See how we've written 8 + 6 = 10 in a slightly different way?
Now here's the thing - we can add 8 + 6 = 10 and we can also add together the 4 and the 3 (the numbers that aren't the common factor) to get 7 and then multiply by 2 (the common factor) to get 14. We can do it either way.
We show that we do the addition first by using brackets:
2 x (4 + 3) = 14
The process of changing an addition-only statement to one that includes multiplication is called Factoring or Factorization.
Vocabulary used:
For more information check out these links (comment to add your favourite link):
Where might you have come from?
Fact-orials Index
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Operations with different kinds of numbers:
Associated Operations:
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