Angol Matematika

Divisibility, divisibility properties, divisibility rules

Definition: if a and b are integers, we say that a is a divisor of b if there is a positive integer q such that aq=b.

The sign aIb can be read in two ways:

a is a divisor of b

b can be divided by a

Properties                                                                                                      

1. aIa -> any integer can be divided by itself

2. if aIb then aIbc -> if a is a divisor of b then a is a divisor of the multiples of b

3. if aIb and bIc then aIc

4. if aIb and aIc then aIb+c ->if a number is a divisor of two different numbers then it is also a divisor of their sum

5. if aIb+c and aIb then aIc -> if a number is a divisor of the sum of two terms and it is also a divisor of one of them then it is also a divisor of the other term

6. if aIb and bIa then a=b

7. if aI1 then a=1

8. for any integer aI0 because 0a=0 (any natural number/integer is a divisor of 0)

Fundamental theorem of arithmetic                                                             

Every positive integer can be written as a product of prime numbers and it can be done in a unique way.

Divisibility rules:                                                                                           

2: the last digit is divisible by 2 (even numbers)

4: the last two digits are divisible by 4

8: the last three digits are divisible by 8

5: the last digit is divisible by 5 (0,5)

25: the last two digits are divisible by 25

125: the last digits are divisible by 125

3: the sum of the digits is divisible by 3

9: the sum of the digits is divisible by 9

7: double the last digit and subtract it from the rest of the number and if the difference is divisible by 7 then the number is also divisible

11: those numbers whose alternating sum of digits (we alternate th signs from positive to negative to positive to negative...) is divisible by 11

Prime numbers                                                                                              

Definition: if a number has only two different factors (divisors), itself and 1, it is called a prime number.

With the exception of 2, every prime number is odd.

Composite number                                                                                        

Definition: a composite number is a positive integer which has a positive divisor other than one or itself.

Any positive integer, which is greater than one and not a prime number, is a composite number.

Relative(ly) primes                                                                                       

Two integers are said to be relative(ly) primes if they share no common positive factors (divisors) except 1. Relative(ly) prime integers are sometimes called strangers or co-primes.

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