Divisibility

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Divisibility . It refers to the possibility that given two integers a and b (in particular two natural numbers) there exists an integer c such that a = bc. If c exists, it will be said that a is divisible by b. If such an event occurs, it is denoted b | a, which reads: b is a divisor of a or b divides the number a . [1] , [2]

In dealing with the divisibility of numbers, nothing is said about the different values ​​it can take when dividing a by b. Let us specify a = bc, and also a = bc 1 that allows us to deduce

bc = bc 1 (a1) or the same b (cc 1 ) = 0, assuming that b is different from 0, c = c 1 . The uniqueness of the quotient is affirmed. However, if b = 0, certainly, that a = 0 and equality a1, it is true for any value of c. So 0 is only divisible by 0, in which case the quotient is undetermined. [3]

Summary

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  • 1 Properties
  • 2 Nomenclature and partition
  • 3 Fundamental principles
  • 4 Divisibility criteria
  • 5 References
  • 6 See also
  • 7 Sources

Properties

  1. Every number is divisible by itself. a | a.
  2. a | b and b | c implies a | c.
  3. If a | b and b | a, then a = b or a = -b.
  4. If b | a | b | > ♀ | a |, then a = 0.
  5. If b | a and different from 0, then | b | <♀ | a |.
  6. So that b | a is necessary and sufficient that | b | divide♀ | a |. [4]

The last property allows us to see only the case in which the divisor is positive. Furthermore the divisibility of any integers is restricted to non-negative integers.

Nomenclature and partition

Focusing on non-negative numbers can be expressed as a prime number , the non-negative integer that has exactly two divisors: the same and 1. For example: 11, 13, 14, 19, etc.

A number that is not prime or 1 is called a composite number . 4, 16, 64, 256, etc.

The natural number 1 is not considered a prime.

In the set of non-negative integers the partition is given that includes the set of the prime numbers, the set of the composite numbers and the unit set formed by 1.

Fundamental principles

The fundamental principles of divisibility are certain properties that serve as a basis to deduce, to meet the conditions that a number must meet to be divisible by another.

  1. If one number divides several others, it divides its sum.

Example : Investigate, without finding it, if the sum of 6; 12 and 15 will be divisible by 3.

Since 3 divides 6, 12, and 15, we can also affirm that the sum of these three numbers will be divisible by 3, even without knowing how much the sum is equal to.

Answer: The sum will be divisible by 3 because 6 is; 12 and 15.

Proof: 6 + 12 + 15 = 33

33: 3 = 11 Checking

  1. If one number divides two others, it divides their difference.

Example : Find out if the difference between 39 and 15 will be divisible by 3 without doing the subtraction.

Since 39 and 15 are multiples of 3, we can assure that their difference will also be a multiple of 3.

Answer: Since they are 39 and 15 divisible by 3, their difference will also be.

III. If one number divides another, it divides any multiple of it.

Example : Investigate, without doing the multiplication, if 14. 4 will be divisible by 7

Since 7 divides 14 we can assure that the product of multiplying 14 by 4 will also be divisible by 7.

Answer: The product will be divisible by 7 because it is 14.

Test: 14. 4 = 56

56: 7 = 8 Checking.

  1. If the addends are not divisible by a number, the sum, however, may be according to the following:

If when dividing given numbers by another number the divisions are not exact, but the sum of the remainders of these divisions is exactly divisible by that other number, the sum of the given numbers will also be divisible by it.

Example : Investigate whether the sum of 23, 53, and 15 will be divisible by 7, without division. We divide each of these numbers by 7 to find the remains.

And as the sum of the remainders 2 + 4 + 1 = 7, the sum of the numbers 23; 53 and 15 will also be a multiple of 7, for sure. Answer: The sum of 23, 53 and 15 will be divisible by 7 because it is the sum of the remains obtained by dividing these numbers by 7.

Test: 23 + 53 + 15 = 91

91: 7 = 13 Checking.

  1. If by dividing two given numbers by a third the divisions are not exact, but the rest of those divisions are equal, the difference of these numbers will be divisible by the third.

Example: Find out without subtracting, if the difference between 26 and 8 will be divisible by 3.

Since the remainders are equal, 2 = 2, we can affirm that the difference between 26 and 8 will be divisible by 3.

Answer: Since dividing 26 and 8 by 3 remains the same, the difference of 26 and 8 will be divisible by 3.

Test: 26 – 8 = 18

18: 3 = 6 checking.

SAW. If a number divides the dividend and the divisor, it divides the rest.

Example : Since 7 divides 21 and 35, the remainder of 35: 21 has to be divisible by 7. Indeed:

The remainder is 14 which is a multiple of 7.

VII. If a number divides the divisor and the rest, it also divides the dividend.

In effect, dividing the divisor also divides the product of the divisor by the quotient according to the third principle.

Divisibility criteria

The following criteria allow us to find out if a number is divisible by another in a simple way, without having to carry out a division:

Number Criterion Example
2 The number ends in zero (0) or even number 476: because “6” is even.
3 The sum of their figures is a multiple of 3 534: because 5+ 3+ 4 = 12 is a multiple of 3
4 The number formed by the last two digits is a multiple of 4 5628: because 28 is a multiple of 4
5 The last number is 0 or 5 560: because it ends in 0
6 The number is divisible by 2 and by 3 432: See previous criteria.
7 For 3-digit numbers: The number formed by the first two digits is subtracted from the last one multiplied by 2. If the result is a multiple of 7, the original number is also a multiple.

For numbers with more than 3 digits: Divide into groups of 3 digits and apply the criteria above to each group. Alternately add and subtract the result obtained in each group and check if the final result is a multiple of 7.

 52176376: because (37-12) – (17-12) + (5-4) = 25-5 + 1 = 21 is a multiple of 7.
8 The number formed by the last three digits is a multiple of 8 42560: because 560 is a multiple of 8
9 The sum of their figures is a multiple of 9 6534: because 6 + 5 + 3 + 4 = 18 is a multiple of 9
10 The last number is 0 920: The last digit is 0
eleven Adding the numbers (of the number) in odd position on the one hand and those of even position on the other. Then the result is subtracted from both sums obtained. If the result is zero (0) or a multiple of 11, the number is divisible by it. If the number has two digits it will be a multiple of 11 if those two digits are the same. If the number has two digits it will be a multiple of 11 if those two digits are the same. 42702: 4 + 7 + 2 = 13 • 2 + 0 = 2 • 13-2 = 11 → 42702 is a multiple of 1166: because the two digits are the same. So 66 is a Multiple of 11
12 The number is divisible by 3 and 4 216: See previous criteria
13 For numbers with more than 3 digits: Divide into groups of 3 digits, alternately add and subtract the groups from right to left and apply the criteria above to the result obtained. If it is a multiple of 13, the original number is also a multiple. 432549: because 549-432 = 117 and then 11 + 4 • 7 = 39 is a multiple of 13.

 

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