   SEARCH HOME Math Central Quandaries & Queries  Question from nazrul, a teacher: If m,n,k are natural number how can I prove that (m+n)k=mk+nk. In the proof the properties of natural number should be used. 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     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.