Hi Nazrul,

I need to write a proof and am not sure where to start, I like to write out the properties I know for the given situation. In this case, the properties of natural numbers:

Closure under addition: If a and b are natural numbers then the sum of a+b is a natural number
Closure under multiplication: If a and b are natural numbers then the product of a×b is a natural number
Associativity under addition: If a, b, and c are all natural numbers then a + (b + c) = (a + b) + c
Associativity under multiplication: If a, b, and c are all natural numbers then a × (b × c) = (a × b) × c.
Commutativity under addition: If a and b are natural numbers then a + b = b + a.
Commutativity under multiplication: If a and b are natural numbers then a × b = b × a
Additive identity: For every natural number a, a + 0 = a and a × 1 = a.
Multiplicative identity: For every natural number a, a × 1 = a.
Distributivity: If a, b, and c are all natural numbers then a × (b + c) = (a × b) + (a × c)

Looking at what you need to prove, I try to recognize any familiar patterns; I know you will have to use the properties commutativity under multiplication and distributivity.

Hope this helps,

Janice