Adding and Subtracting Multiples
If you add two multiples of 2 together, what is true about the sum (Hint: a multiple of 2 is an even number)?
That's correct.
If you think of any two even numbers and add them together, the sum will always be even.
Not so.
Think of any two even numbers and add them together, and you'll see the number is always even.
Next, add a multiple of 2 with an integer that isn't a multiple of 2. What is true about the sum?
Not so.
Think of an even number and an odd number and add them together and see what you get. Any time you add add a multiple of 2 with an integer that isn't a multiple of 2, the result is never a multiple of 2.
That's right!
Any time you add add a multiple of 2 with an integer that isn't a multiple of 2, the result is never a multiple of 2.
The last possibility to consider is if you add or subtract two numbers that are not multiples of _n_.
Suppose you add together two numbers that are not multiples of 3. What is true about the sum?
Not quite.
It is sometimes the case that you can find two integers that aren't multiples of 3 that add up to a multiple of 3 (e.g., 4 + 5 = 9). Still, there are also examples that don't add up to multiples of 3 (e.g., 4 + 7 = 11).
Not quite.
It is sometimes the case that you can find two integers that aren't multiples of 3 that don't add up to a multiple of 3 (e.g., 4 + 7 = 11). Still, there are also examples that do add up to multiples of 3 (e.g., 4 + 5 = 9).
Exactly!
You can find examples of integers that aren't multiples of 3 that do add up to a multiple of 3 and that don't add up to a multiple of 3 (e.g., 4 + 5 = 9 and 4 + 7 = 11).
Putting all that information together:
[[summary]]
For example, 4 and 8 are multiples of 2 (even numbers), and their sum, 12, is also a multiple of 2 (even). This will always be the case. In fact, this is the case for multiples of any number, not just 2. For any integer, _n_, the sum (or difference) of two multiples of _n_ will also be a multiple of _n_.
For example, 6 is a multiple of 2 (even), and 7 is not a multiple 2 (odd). Their sum is 13, which is not a multiple of 2 (odd).
This conclusion also applies to multiples of any number, not just 2. For any integer, _n_, the sum (or difference) of a multiple of _n_ with a non-multiple of _n_ will never be a multiple of _n_.
The sum is a multiple of 2 (even).
The sum is not a multiple of 2 (odd).
The sum could be a multiple of 2 or not.
The sum is a multiple of 2 (even).
The sum is not a multiple of 2 (odd).
The sum could be a multiple of 2 or not.
The sum is a multiple of 3.
The sum is not a multiple of 3.
The sum may or may not be a multiple of 3.
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