Divisibility theory

Divisibility

 

Divisibility is the property of an integer number to be divided by another, resulting an integer number.

Where a and b, two integers numbers, we will say that “a” is a multiple of “b” if there is an integer “c”, such as, when we multiply it by “b” is equal to “a”. That is, “a” is a multiple of “b” if:

bioprofe |exams with exercises about physics, chemistry and mathematics | Divisibility

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Being “c” another integer number.

The set of all multiples of an integer “a” is obtained by multiplying “a” for each integer number.

Example:

We write the set of multiples of 6:

bioprofe |exams with exercises about physics, chemistry and mathematics | Divisibility

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  • The set of multiples of (-7):
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Similarly, “b” is divisor of “a” if the division of “a” between “b” is an integer number.

If “a” is an integer number, it is always fulfilled that 1 and -1 are divisors of “a”, as the divisions:

bioprofe |exams with exercises about physics, chemistry and mathematics | Divisibility

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Are integer numbers:

bioprofe |exams with exercises about physics, chemistry and mathematics | Divisibility

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The set of divisors of an integer “a” is symbolized by: Div (a), and in practice, it is obtained by calculating the positive divisors of “a” and adding their opposites.

Example:

  • We write the set of the dividers of -18:
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Divisibility criteria:

  • A number is divisible by 2 when it is odd or ends in 0, 2, 4, 6, or 8.
  • A number is divisible by 3 if the sum of its digits is a multiple of 3.
  • A number is divisible by 4 when its last two digits are zeros or they are a multiple of 4.
  • A number is divisible by 5 when it ends in 0 or 5.
  • A number is divisible by 6 when it is divisible by 2 and by 3 simultaneously.
  • A number is divisible by 7 when separating the first digit on the right, multiplying by 2, subtracting this product from what is left and so on, gives zero or a multiple of 7.
  • A number is divisible by 8 when its last three digits are zeros or they are a multiple of 8.
  • A number is divisible by 9 when the sum of its digits is a multiple of 9.
  • A number is divisible by 10 when it ends in 0.
  • A number is divisible by 11 when the difference between the sum of the absolute values of odd numbers place and the sum of the absolute values of torque figures place from right to left, is zero or a multiple of 11.
  • A number is divisible by 13 when separating the first digit on the right, multiplied by 9 subtracting this product from what is left and so on, gives zero or a multiple of 13.
  • A number is divisible by 17 when separating the first digit on the right, multiplied by 5, subtracting this product from what is left and so on, gives zero or a multiple of 17.
  • A number is divisible by 19 when separating the first digit on the right, multiplied by 17, subtracting this product from what is left and so on, gives zero or multiple of 19.
  • A number is divisible by 25 when its last two digits are zeros or they are a multiple of 25.
  • A number is divisible by 125 when its last three digits are zeros or are a multiple of 125.

Decomposition of an integer number into prime factors:
An integer number is prime if its only divisors are itself, the unity and their opposites.

Example:

  • We check that 13 is a prime number:
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It is called prime factorization of an integer number to the process that allows us to write that number as a product whose factors are prime numbers.
In practice, to decompose a positive number entered into prime factors, it is written and, on the right, the first prime number greater to one which it is divisible. The division is performed and the quotient is written to the left, below the initial number. These steps are repeated until the quotient obtained is 1.

 

Example:

  • Decompose the integer number 180 into prime factors:
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To break a negative integer into prime factors, we will leave aside the minus sign and we multiply by -1 the decomposition obtained.

 

Greatest common divisor of two or more integers:
We will call common divisor of a and b and we denote it by G.C.D. (A, b) to the higher of the common divisors of a and b.
If the G.C.D. (a, b) is 1, it is said that the numbers a and b are prime numbers between them.
To find the G.C.D (a, b) we should proceed as follows:

 

  • If any of them is negative, we dispenses with its sign.
  • Decomposition is performed on each of prime numbers obtained in the preceding paragraph factors.
  • The greatest common divisor is the product of high common prime factors, each one at the lowest exponent.

Example:

  • Calculate the greatest common divisor of 1800 and 990: We decompose both numbers into prime factors
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    Least common multiple of two or more integers.
    We will call least common multiple of “a” and “b” and we represent it as l.c.m. (a, b) to the lowest common positive multiples of a and b.
    To find the l.c.m. (a, b) we have to proceed as follows:

    • If any of them is negative, we dispenses of its sign.
    • Decomposition is performed on each of prime numbers obtained in the preceding paragraph factors.
    • The least common multiple is the product of prime factors common and uncommon high, each of them, elevated to the greatest exponent.

     

    Example:

    We calculate the least common multiple of 600 and 495: both numbers are decomposed into prime factors:

bioprofe |exams with exercises about physics, chemistry and mathematics | Divisibility

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Divisibility

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