# Multiplying Integers

The multiplication of several integers is another integer. The absolute value of the product is the multiplication of the absolute values of the factors, and the sign of the product can be determined by using the rule of signs: 2 · 5 = 10

(−2) · (−5) = 10

2 · (5) = −10

(−2) · 5 = −10

### Properties of the Multiplication of Integers

1. Closure:

The result of multiplying two integers is another integer.

a · b  2 · (−5)  2. Associative:

The way in which the factors are grouped does not change the result.

(a · b) · c = a · (b · c)

(2 · 3) · (−5) = 2· [(3 · (−5)]

6 · (−5) = 2 · (−15)

−30 = −30

3. Commutative:

The order of the factors does not change the product.

a · b = b · a

2 · (−5) = (−5) · 2

−10 = −10

4. Multiplicative Identity:

The 1 is the neutral element of the multiplication because any number multiplied by it gives the same number.

a · 1 = a

(−5) · 1 = (−5)

5. Distributive:

The product of a number for a sum is equal to the sum of the products of this number for each of the addends.

a · (b + c) = a · b + a · c

(−2) · (3 + 5) = (−2) · 3 + (−2) · 5

(−2) · 8 = (−6) + (−10)

−16 = −16

Removing a common factor:

It is the reverse of the distributive property.

a · b + a · c = a · (b + c)

(−2) · 3 + (−2) · 5 = (−2) · (3 + 5)