### Properties of Multiplication on Integers:

**Property 1: (Closure Property)**

The product of two integers is always an integer.

**Verification:** We have,

1) 6 x 5 = 30, 30 is an integer

2) 4 x (-5) = -20, -20 is an integer

3) (-9) x (8) = -72, -72 is an integer

2) (-3) x (-2) = 6, 6 is an integer

**Property 2: (Commutative Law for Multiplication)**

For any two integers a and b, we have **a x b = b x a**.

**Verification:** We have,

5 x (-7) = -35 and (-7) x 5 = -35

Therefore, 5 x (-7) = (-7) x 5.

**Property 3:** **(Associative Law for Multiplication)**

If a, b, c are any three integers then **(a x b) x c = a x (b x c)**.

**Verification:** Consider the Integers -2, —4 and -6.

We have,

((—2) x (—4)) x (-6) = -48.

And, (—2) x ((—4) x (-6)) = (-2) x (24) = —48.

Therefore, ((—2) x (—4)) x (-6) = (—2) x ((—4) x (-6))

**Property 4:** **(Distributive Law for Multiplication)**

If a, b, c are any three integers then **a x (b + c) = a x b + a x c**.

**Verification:** Consider the Integers 7, —5 and -2.

We have,

7 x ((—2) + (—4)) = 7 x (-6) = -42.

And, 7 x (—2) + 7 x (-4) = (-14) + (-28) = —42.

Therefore, 7 x ((—2) + (—4)) = 7 x (—2) + 7 x (-4)

**Property 5:**

For any integer a, we have a x 1 = a. The integer **1** is called **multiplicative identity** for integers.

**Verification:** We have,

1) (-6) x 1 = -6

2) 5 x 1 = 5

**Property 6:**

For any integer a, we have a x 0 = 0.

**Verification:** We have,

1) 7 x 0 = 0

2) 5 x 0 = 0