Discrete Math Proof: Necessary Condition for Divisibility by 6

[in the rye]

Homework Statement

We have JUST started writing proofs recently, and I am a little bit doubtful in my abilities in doing this, so I just want to verify that my proof actually works. I was expecting this one to be a lot longer since the previous 2 were. I don't see any glaring flaws in it, but I'd just like to be sure (writing these feel awkward since this is my first proof base course):

A necessary condition for an integer to be divisible by 6 is that it is divisible by 2.

Homework Equations

The Attempt at a Solution

Assume true.

Pf./
[For all integers n, if n is divisible by 6, then n is divisible by 2]. Assume 6|n, n ∈ ℤ. By definition, n = 6k, k ∈ ℤ. Consider that n = 2(3k). See that 3k ∈ ℤ has closure by multiplication of the set of integers, and let 3k = t, t ∈ ℤ. Notice n is even since n = (3k) = 2t. Therefore, 2|n. QED.

 

[Mark44]

Homework Statement

We have JUST started writing proofs recently, and I am a little bit doubtful in my abilities in doing this, so I just want to verify that my proof actually works. I was expecting this one to be a lot longer since the previous 2 were. I don't see any glaring flaws in it, but I'd just like to be sure (writing these feel awkward since this is my first proof base course):

A necessary condition for an integer to be divisible by 6 is that it is divisible by 2.

Homework Equations

The Attempt at a Solution

Assume true.

Pf./
[For all integers n, if n is divisible by 6, then n is divisible by 2]. Assume 6|n, n ∈ ℤ. By definition, n = 6k, k ∈ ℤ.

n = 6k = 2*3k, so n is even, hence is divisible by 2. That's really all you need to say.

Consider that n = 2(3k). See that 3k ∈ ℤ has closure by multiplication of the set of integers, and let 3k = t, t ∈ ℤ. Notice n is even since n = (3k) = 2t. Therefore, 2|n. QED.

 

[in the rye]

n = 6k = 2*3k, so n is even, hence is divisible by 2. That's really all you need to say.

Thanks. I'm having trouble being too wordy because I feel like I need to cover my basis. Do you have any general tips to proof reading (no pun intended) proofs?

 

[Mark44]

Thanks. I'm having trouble being too wordy because I feel like I need to cover my basis. Do you have any general tips to proof reading (no pun intended) proofs?

"Keep it as simple as possible, but no simpler." I don't know how much help that is, but if you have trouble being too wordy (as you said), see if you have extra baggage in there that isn't needed. In you proof you have this sentence: "See that 3k ∈ ℤ has closure by multiplication of the set of integers, and let 3k = t, t ∈ ℤ." It's completely unnecessary.