n(n−1) is even why ? It is divisible by 2 but why it's an intefer ?

[spice lark]

n(n−1)​/2=Even​

[white vector]

let n be an odd number what would n-1 be

[spice lark]

I get that why we assume is a integer ? for a ration (terminating decimal)

[white vector]

uhhh what

btw n(n−1)​=even

not n(n−1)​/2= even

it's just that in any 2 adjacent numbers one of them has to be even

so their product is even

[long hedge]

You know the definition of divisibility, right?

[fickle cypress]

We're not assuming an integer, this is only valid for an integer

[long hedge]

Fr tho
If you know that n(n - 1) is an even number
that means there exists an INTEGER x such that n(n - 1) = 2x
and 2x/2 = x, which is, by existence, an INTEGER

[fickle cypress]

Yea ofc

[tall sage]

n and n - 1 are neighbors of each other

and because elements of the set {n|n\equiv 0\mod 2}\cup{n|n\equiv 1\mod 2}=Z such that both of them loop in binary we can say that there are two cases

(1) n is in the first set (it's even)

that implies n(n - 1) is a multiple of an even integer

hence it's even

(2) n is in the second set (it's odd)

[errant owl]

dont use latex

[tall sage]

that implies n - 1 no longer has remainder 1 and will be an even integer

hence n(n - 1) is also a multiple of an even integer

hence it's also even

[tall sage]

you know what i'm going to omit the Z in the second part

i think it's implied that n in Z anyways

[fervent python]

ghost

[delicate flax]

Alternatively, if you have $n$ people and need to make a line of $n-2$, there's $\frac{n(n-1)}2$ ways to do that, so $\frac{n(n-1)}2$ is integral and $n(n-1)$ is even

[fervent python]

BlockLayer2000

[obtuse wing]

i fear humanity

[charred nest]

because an even number has a factor of 2 and any number with a factor of 2 is even so 2 times an odd is even and so is an even times an even

[heady night]

n typically refers to only integer values. thus this looks something like integer(integer-1)/2

since an integer, times itself minus 1 is always even

you will divide it by two and receive an integer

does that make sense?

This is because an even times an odd always results in an even number