This is the second post in the Divisibility Rules Series. In the last post, we discussed about divisibility by 2. In this post, we discuss divisibility by 4.
Now, how do we know if a number is divisible by ?
Four divides because . It is also clear that four divides and all multiples of . Therefore, four divides multiples of , , and . In general, divides , where is an integer greater than .
Now, how do we know if a number that is not a power of is divisible by . Let us try a few examples.
Example 1: Is divisible by ? is equal to and is divisible by . Since is also divisible by , therefore, id divisible by .
Example 2: Is divisible by ? is equal to . Now, is divisible by . Since is not divisible by , therefore, is not divisible by .
Example 3: Is divisible by ? . Now, is divisible by (it’s a multiple of 100), and is not divisible by . Therefore, is not divisible by .
By now, you would have realized that we just test the last 2 digits of the numbers if we want to find out if it is divisible by 4: 148, 362, and 3426. » Read more
The Divisibility Rules Series. In this series, we discuss the different divisibility rules. We discuss the reasons and proofs behind the rules.
- Divisibility by 2
- Divisibility by 4
- Divisibility by 5 and 10
- Divisibility by 3
- Divisibility by 6
- Divisibility by 7
- Divisibility by 8
- Divisibility by 9
- Divisibility by 11
- Divisibility by 12
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.