# Multiplication of Natural Numbers

To multiply two natural numbers, add one factor to itself many times as indicated by the other factor.

**a · b = c **

The terms** a and b** are calle** factors and the result, c, is the**** product. **

### Properties of the Product of Natural Numbers

1. **Closure**:

The result of multiplying** two natural numbers** is another** natural number**.

**a · b **

**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**

3 · 1 = 3

**5. Distributive**:

The multiplication of a natural number and a sum is equal to the sum of the** multiplication** of the** natural 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)

6 + 10 = 2 · 8

16 = 16