Question

Is the sum of a multiple of 3 and a multiple of 3 closed under addition ?

Answer 1

Answer:

Yes, the sum of a multiple of 3 and a multiple of 3 is closed under addition.

Step-by-step explanation:

Let S = { 3n | n ∈ Z } be the set of multiples of 3.

So if x and y are two multiples of 3, then we can write that

x = 3n and y = 3m for some integers n and m in the set of integers i.e. Z.

To determine that the sum of 3n and 3m is closed under addition. As

x + y = 3n + 3m = 3(n + m) ∴ from the distributive property

As we can check that n + m is also an integer, as n + m ∈ Z  

Also, x + y = 3 (n + m) is a multiple of 3.

Therefor, the sum of a multiple of 3 and a multiple of 3 is closed under addition.  

Keywords: closure property under addition, multiple of 3

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