What are the divisibility criteria and what are they for?

The divisibility criteria are used to determine whether or not a number is divisible. by another without having to perform the division. In unProfesor we tell you with easy examples to study.

In a new lesson from a Teacher we are going to study What are the divisibility criteria and what are they for?. First we will talk about their importance, what they are for and what they are. Then we will continue with the criteria from numbers 2 to 10 and we will finish with the decomposition into prime numbers.

Index

1. What are the divisibility criteria?
2. What are the divisibility criteria used for?
3. The divisibility criteria from 2 to 10
4. What is prime factorization?

Divisibility criteria are a set of rules separated by numbers, that enables anyone who knows them to know in a simple way if a number is divisible by another. By this we mean that A number will be divisible by another if it meets the divisibility criterion. of that number and the result of the division is zero. If the remainder is non-zero, then the number is NOT divisible.

The criteria are used to find out WITHOUT having to do the division, whether or not one number is divisible by another. If the numbers we are trying to figure out are very small, perhaps we can use the tables to remember whether or not they are multiples and therefore divisors, but when the numbers are very large, it is very useful to know the criteria of divisibility.

Here we indicate what are the divisibility criteria.

Now that you know what they are, we are going to tell you What are the divisibility criteria used for? Here we indicate it:

• They are useful for finding the divisors of any number.
• They allow you to decompose a number into prime factors.
• It makes it possible to know if a number is prime or composite.
• It helps when simplifying fractions.

What is a multiple and what is a divisor?

The multiples of a number are those natural numbers that result from multiplying that number by the natural numbers. And the divisors are those natural numbers whose division with another number gives an exact result, that is, the remainder is zero.

Discover what are the divisors of a number.

Next, the divisibility criteria from number 2 to number 10.

• criterion of 2: All even numbers are divisible by 2. Example: 28 is divisible by 2 because it ends in 8, an even number. 28 / 2 = 14.
• Criterion 3: All numbers whose sum of their digits is equal to three or a multiple of three are divisible by 3. Example: 15 is divisible by 3, because 1 + 5 = 6 which is a multiple of 3. 15 / 3 = 5.
• Criterion 4: All numbers whose last two digits are zero or multiples of four are divisible by 4. Example: 128 is divisible by 4 because 28 is a multiple of 4. 128 / 4 = 32.
• Criterion 5: All numbers whose last digit is equal to zero or five are divisible by 5. Example: 135 is divisible by 5 because it ends in 5. 135 / 5 = 27.
• Criterion 6: All numbers that meet the divisibility criteria of 2 and 3 at the same time are divisible by 6. That is, it must be divisible by both numbers. Example: 90 is divisible by 6, because it is divisible by 2 because it is even, and it is divisible by 3 because the sum of its digits is 9, which is a multiple of 3. 90 / 6 = 15.
• Criterion 7: are divisible by 7 all numbers whose last digit multiplied by two and subtracted from the number formed with the remaining digits, and repeating the process until a single digit number is obtained and it is seven or zero. Example: 35 is divisible by 7 because 5 x 2 = 10, 10 - 3 = 7. 35 / 7 = 5. In unProfesor we discover the divisibility of 7.
• criterion of 8: all numbers whose last three digits are multiples of eight or all zero are divisible by 8. Example: 2000 is divisible by 8 since its last three digits are zero. 2000 / 8 = 250
• Criterion 9: All numbers whose sum of their digits is a multiple of nine are divisible by 9. Example: 81 is divisible by 9 because 8 + 1 = 9. 81 / 9 = 9.
• Criterion of 10: All numbers ending in zero are divisible by 10. Example: 130 is divisible by 10 because it ends in zero. 130 / 10 = 13

The prime factorization consists in divide a number into its prime divisors until finished only with number 1. In this way, we can see what the divisors of a number are and remember what the prime numbers are.

The Prime numbers are those who can only divide by one and by themselves. And composite numbers are those that, in addition to being divided by one and by themselves, are divisible by other numbers.

For example: we want to factor the number 420 into prime numbers.

We start first with number 2.

420 / 2 = 210

We divide by 2 again.

210 / 2 = 105

Since the number 105 is not an even number, we know from the divisibility test for 2 that it is not divisible. Therefore we begin to divide by the next prime number which is 3.

105 / 3 = 35.

As the number 35 does not add 3 or a multiple of 3 with its digits, since 3 + 5 = 8. It is not divisible by 3. So we continue to divide it by the next prime number which is 5.

35 / 5 = 7

The number 7 is a prime number, therefore it will not be divisible by 5. It will only be divisible by itself.

7 / 7 = 1.

By obtaining the number 1 as a result, we finish the decomposition of the number 420.

We can then write the number 420 as follows:

420 = 2 x 2 x 3 x 5 x 7

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