Question

Number Proof Help: For an arbitrary integer a, verify the following:

2 | a(a+1), and 3 | a(a+1)(a+2)

Answer 1

pretty clear that if you have three consecutive integers, that 3 has to divide one of them

Answer 2

... WOW I feel pretty dumb right now. Thanks for showing me the general thought process

Answer 3

But for 2 divisible by a(a+1) if a=-1 would that make it so that 2 is divisible by 0? Is that true?

Answer 4

Nope, but that 0 is divisible by 2, so is the meaning of the notation in the above question.

Answer 5

Oh I see thanks, yeah I keep forgetting that its only when something is divided by 0, but anyway how would I write a proof for this?

I know that 2 divides into all even numbers, and that a(a+1) will always be even since
if a is odd then a+1 is even and any number multiplied by an even number is even.
if a is even then the same concept applies since any number multiplied by an even number is even. So how would I state this as verifying it?

its a similar concept with the other question but I can answer that on my own if I can get a general view on how I can answer the first one.

Answer 6

The thought process you had for the first question is correct. You can write the proof as is (with little revision) or use notations like the above.

Let a be even. That is, a = 2k, where k is an integer. Then a + 1 = 2k + 1, and so it is odd. since 2 | a, then 2 | a(a + 1).

Let a = 2k + 1 so that it is odd. Then a + 1 = (2k + 1) + 1 = 2k + 2 = 2(k + 1), and so it is even. Since 2 | (a + 1), then 2 | a(a + 1).

For the last question, you can use the same method above.

Let a be divisible by 3. (a = 3k, k an integer). Then check.
Let a leaves a remainder of 1 when divided by 3. (a = 3k + 1). Then check.
Let a leaves a remainder of 2 when divided by 3. (a = 3k + 2). Then check.

Answer 7

thank you

Answer 8

Your welcome. :)