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The Divisibility Rules Series. In this series, we discuss the different divisibility rules. We discuss the reasons and proofs behind the rules.
How do we know if a number is divisible by a certain number?
In this post, the first post in the Divisibility Rules Series, we examine why a number is divisible by 2 if it is even. In this post, since we are talking about divisibility rules, when we use the word number, we mean integer.
Since multiplication and division are inverses of each other, we can examine what happens if a number is multiplied by 2. Let’s try a few examples:
0 x 2 = 0
1 x 2 = 2
2 x 2 = 4
3 x 2 = 6
4 x 2 = 8
5 x 2 = 10
6 x 2 = 12
7 x 2 = 14
8 x 2 = 16
9 x 2 = 18
From the list above, we make the following observations: (1) the ones digit of numbers multiplied by 2 is either 0, 2, 4, 6, or 8; and (2) if the numbers are consecutive the pattern repeats.
Since we have exhausted all 1-digit numbers in the list above, it is clear that the ones digit of a number multiplied by 2 cannot be 1, 3, 5, 7 or 9. Therefore, we can conclude that a number is divisible 2 if its ones digit is even.
When I was still teaching, several of my students asked me this: Is 0 a natural number?
Short Answer
It depends on the convention. In some references, the set of natural numbers includes 0. In other references, the set of natural numbers does not include 0.
Long Answer
When we define a particular mathematical concept, we come to some sort-of agreement. For example, we might want to call the sequence 1, 4, 7, 10, … “cool numbers.” If all of us agree on this, we will have no problem. However, other group of uncool guys might give it another name. Or, they might keep the name, but change the sequence. This was what happened to the set of natural numbers. » Read more