Properties of Divisibility of Integers

A division problem has four main parts: the dividend (the number being divided), divisor (the number doing the dividing), quotient (the answer), and remainder (what is left over).

 

For example, in 10 ÷ 3 = 3 R1:


1. Reflexive property

       Any number divided by itself equals 1. If a is an integer, then it is written as;

a | a 

It means that a is divisible by a.

 

Examples:

 

2. Divisibility by 1

         Any number divided by 1 is still the same number, and division by 1 always gives an integer, then it is written as

1 | a

for all integers a, this means 1 is a divisor of a.

 

Examples:

 

3. Transitive property

       If one number divides a second, and the second divides a third, then the first divides the third, then it is written as

If a | b and b | c then a | c

If a divides b and b divides c, then a divides c.

 

Example 1:

2 | 4, 4 | 8

a = 2, b = 4, c = 8

If a | b and b | c then a | c

then 2 | 8

2 | 4, 4 | 8, then 2 | 8

 

Example 2:

3 | 6, 6 | 18

a = 3, b = 6, c = 18

If a | b and b | c, then a | c

then 3 | 18

3 | 6, 6 | 18, then 3 | 18

 

Example 3:

5 | 10, 10 | 20

a = 5, b = 10, c = 20

If a | b and b | c, then a | c

then 5 | 20

5 | 10, 10 | 20, then 5 | 20

 

4. Addition and subtraction property

       If a number divides two integers, it also divides their sum and difference, then it is written as

If a | b and a | c, then a | (b + c), a | (b - c)

If a divides b and a divides c, then a divides b + c and b − c.

 

Example 1:

3 | 9, 3 | 12,

a = 3, b = 9, c = 12

If a | b and a | c, then a | (b + c), a | (b - c)

then 3 | (9 + 12), 3 | (9 - 12)

then 3 | 21,     3 | -3

3 | 9, 3 | 12, then 3 | 21, 3 | −3

 

Example 2:

4 | 8, 4 | 16

a = 4, b = 8, c = 16

If a | b and a | c, then a | (b + c), a | (b - c)

then 4 | (8 + 16), 4 | (8 - 16)

then 4 | 24, 4 | -8

4 | 8, 4 | 16, then 4 | 24, 4 | -8

 

Example 3:

6 | 12, 6 | 18

a = 6, b = 12, c = 18

If a | b and a | c, then a | (b + c), a | (b - c)

then 6 | (12 + 18), 6 | (12 - 18)

then 6 | 30, 6 | -6

6 | 12, 6 | 18, then 6 | 30, 6 | -6

 

5. Multiplication property

       If an integer divides another integer, it will also divide that integer multiplied by any other integer, then it is written as

 If a | b, then a | bc

If a divides b, then a also divides b times c. Where c is any integer you choose

 

Example 1:

3 | 6

a = 3, b = 6

If a | b, then a | bc

If c = 4

then 3 | (6)(4)

then 3 | 24

3 | 6, then 3 | 24

 

Example 2:

5 | 10

a = 5, b = 10

If a | b, then a | bc

If c = 3

then 5 | (10)(3)

then 5 | 30

5 | 10, then 5 | 30

 

Example 3:

4 | 12

a = 4, b = 12

If a | b, then a | bc

If c = 5

then 4 | (12)(5)

then 4 | 60

4 | 12, then 4 | 60

 

6. Zero property

       Zero is divisible by every nonzero integer because zero divided by any nonzero integer equals zero, then it is written as

If a | 0 for all a ≠ 0

If a divides 0 for all nonzero a.

 

Examples:

 

7. Absolute value property

       Divisibility depends on the size of numbers, not their signs. If one number divides another, their absolute values will also divide each other,

then it is written as

If a | b  ⟺  |a| | |b|

If a divides b if and only if the absolute value of a divides the absolute value of b.

 

Examples:

 

8. Comparison property

       If one integer divides another (and the second is not zero), the divisor cannot be larger in absolute value than the number being divided, then it is written as

If a | b and b ≠ 0, then |a| ≤ |b|

If a divides b and b is not zero, then the absolute value of a is less than or equal to the absolute value of b.

 

Examples: