Divisibility rules

Now I’m going to explain what this is about divisibility rules. You will learn to know if a number is divisible among other without the need to make the division.

But what does it mean that one number is divisible among another?

You encounter the problem that in order to know if the division is exact, you have to do it. You know that there are some divisibility rules, but you don’t know how they apply.

Test with 2 at the end of the division, you see that it does not give exact. Then you divide it by 3 and neither. You continue with the 4 and so on until you find the 5 and then yes, you have found a number. But you’ve been operating for a while now and you’re getting tired.

Dividing by 5, the result is 109.317. Another high number, back to square one… you leave it as impossible…

Does this situation ring a bell?

Then on this page you will learn how to use the divisibility rules, or also called divisibility criteria.

To begin with, what does it mean that one number is divisible by another? Let’s see it

What is divisibility in mathematics

It is said that a number is divisible by another number when the result of its division is exact, that is, when its remainder is 0, or in other words, when the result is a natural number.

For one number to be divisible among another is to ask: can it be divided by this number? And when it is asked if it can be divided, it means if the result is going to be exact.

Let’s see it with an example:

Is 10 divisible by 2?

Then we have to ask ourselves, can the 10 be divided by 2?

When we make the division, we see that its result is exact. It is a natural number:

which are the rules of divisibility

Therefore, the 10 is divisible by 2.

Another example: is 10 divisible by 3?

We ask ourselves again: can the 10 be divided by 3?

divisional rules from 1 to 10

The result is not exact and therefore the 10 is not divisible by 3. (cannot be divided by 3)

Get it? To see if it is divisible is to check if when dividing, the result is exact.

All the numbers by which it can be divided are divisors of that number.

By definition, all numbers are divisible to at least 1 and to each other. When they are not divisible by any other numbers, we are talking about prime numbers.

If, in addition to being able to be divided by 1 and between themselves, it can be divided by some other divisor, which makes the rest 0, then it is a compound number.

Divisibility rules. Examples.

There are a number of rules called Divisibility Rules or Divisibility Criteria that let you know if a number is divisible by another, without the need to make the division, which is very useful especially in large numbers.

Generally, these rules are used to break down a number into prime factors.

The most important are the rules of the first prime numbers: 2, 3 and 5

Divisibility rule of 2

A number can be divided by 2, when it ends in 0 or par (2, 4, 6 and 8).
.

To remember this rule, note that 2 is also an even number.

Let’s see an example of how to apply it:

Is the number 26 divisible by 2?

Yes, because it ends in 6 and 6 which is even.

Is the number 6548 divisible by 2?

Yes, because it ends in 8 and 8 is even

The number 547, is it divisible by 2?

No, because it ends in 7, which is odd.

And so on and so forth. Easy, isn’t it?

Divisibility rule of 3

A number can be divided by 3, when the sum of its digits is a multiple of 3.

This may be the most complicated rule of the 3. Let’s look at it more slowly.

To apply it you have to add the digits of the number and check if the sum is a multiple of 3. If so, it will be divisible by 3.

Is 12 divisible by 3?

We have to add its figures: 1+2 = 3

3 is a multiple of 3, therefore 12 is divisible by 3

42, is it a multiple of 3?

The sum of its figures is: 4+2 = 6

Is 6 a multiple of 3? Yes, therefore 42 is divisible by 3

246, is it divisible by 3?

The sum of its figures is: 2+4+6 = 12

12 is a multiple of 3, then 246 is divisible by 3.

5465, is it divisible by 3?

The sum of its figures is: 5+4+6+5 = 20

20 is not a multiple of 3, then 5465 is not divisible by 3.

Divisibility rule of 3

This rule is also very easy.

A number can be divided by 5, when it ends in 0 or 5.

Let’s look at a few examples:

5475 Is it divisible by 5?

Yes, because it ends in 5

45780 is it divisible by 5?

Yes, because it ends in 0

34681 is it divisible by 5?

No, because it doesn’t end in 0 or 5.

A number can be divisible by more than one number. For example 30:

  • It is divisible by 2, because it ends in 0
  • It is divisible by 3, because the sum of its figures is 3, which is a multiple of 3
  • It is divisible by 5, because it ends in 0

Divisibility rules for other prime numbers

In practice, divisibility rules will be used when you have to break a number down into prime factors, which we’ll see in the next lesson.

Knowing the three previous rules, you will be able to break down the vast majority of numbers.

I honestly don’t know any other.

The rules of divisibility of the other prime factors are not at all practical, since less time is spent in making the division than in applying the rule. Therefore, I recommend to make the division directly, since you don’t have to memorize any rule.

For example, the divisibility rule of 7 is: A number is divisible by 7 when the difference between the number without units and the double of the units is 0 or a multiple of 7. In this case, it takes less time to divide by 7 and check if it is exact or not.

Finally, remember that these rules only exist for prime numbers. That is, there is no divisibility rule of 4 or 6, since these numbers are composed.

Exercises on divisibility rules

1 – Indicate if the numbers 24, 58, 61 and 586 are divisible by 2. Justify your answer

2 – Indicate if the numbers 24, 336, 651 and 472 are divisible by 3. Justify your answer

3 – Indicate if the numbers 35, 567, 720 and 100 are divisible by 5. Justify your answer

4 – Indicate if the numbers 240, 654, 6571 and 16585 are divisible between 2, 3 and 5. Justify your answer

Solution

1 –

24, yes it is divisible by 2 because it ends in 4, which is par

58, if it is divisible by 2 because it ends in 8, which is par

61, is not divisible by 2 because it ends in 1, which is odd

586, yes it is divisible by 2 because it ends in 6 which is par

2 –

24

2+4 = 6

6 is a multiple of 3

24 if divisible by 3

336

3+3+6 = 12

12 is a multiple of 3

336 yes is divisible by 3

651

6+5+1 = 12

12 is a multiple of 3

651 yes is divisible by 3

472

4+7+2 = 13

13 is not a multiple of 3

472 is not divisible by 3

3 –

35, yes it is divisible by 5 because it ends in 5

567, is not divisible by 5 because it ends in 7

720, yes it is divisible by 5 because it ends in 0

100, yes it is divisible by 5 because it ends in 0

4 –

240

  • Yes is divisible by 2 because it ends in 0
  • The sum of its figures is 2+4 = 6, which is a multiple of 3. 240 is divisible by 3.
  • Yes is divisible by 5 because it ends in 0

654

  • Yes is divisible by 2 because it ends in 4
  • The sum of its figures is 6+5+4 = 15, which is a multiple of 3. 654 is divisible by 3.
  • It is not divisible by 5 because it ends in 4

6571

  • It is not divisible by 2 because it ends in 1
  • The sum of its figures is 6+5+7+1 = 19, which is not a multiple of 3. 6571 is not divisible by 3.
  • It is not divisible by 5 because it ends in 1

16585

  • It is not divisible by 2 because it ends in 5
  • The sum of its figures is 1+6+5+8+5 = 25, which is not a multiple of 3. 16585 is not divisible by 3.
  • Yes is divisible by 5 because it ends in 5

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