Simple divisibility equation

[golden rock]

What's the algebra to prove that if x is divisible by x-1 then x=2 or x=0?

[latent lotus]

how can x be 1?
that will result in divisioon by 0

[rich yarrow]

x-1|x so x/x-1 is an integer

split the fraction as (x-1+1)/(x-1) => x-1/x-1 +1/x-1

1+ 1/x-1 = integer so 1/x-1 is an integer

[golden rock]

Silly mistake on my part

[golden rock]

And how do you solve 1/x-1 is an integer ith algebra?

[rich yarrow]

for 1/x-1 to be an integer, x-1|1 which is possible if |x-1| = 1 so x= 2 or 0

[golden rock]

Ok ok

[zealous rose]

you could think of it this way

x is a neighbor of x - 1

which means certainly gcd(x, x - 1) = 1

or they're coprime

they can't divide each other except for very trivial solutions

that is you could do this

x - 1 = +-1

1 divides x in Z

and so does -1

want you want is the value of x such that x - 1 is equal to either

so you solve for x and you get

x = 0, 2

[cobalt tangle]

nah ignore that

[golden rock]

.close