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# number theory and divisibility watch

1. I'm working through this book with this, and it says the following roughly:

If m and n are integers, and n divides m, then n is a factor of m and m is a multiple of n. therefore m = kn, where k is also an integer, and when you multiply the factor n by k, you get m. therefore k = m/n.

I understand this part

But it tells me to solve divisibility problems you should never use m/n, since m and n are integers so m/n is confusing or something. you have to use m = kn. it asks some questions:

x is a multiple of x^2 + 1. give the integer values of x.

so I did like it asked, and wrote x^2 + 1 = kx

It then gave a hint to rearrange this equation to make is so two integers multiplied gives 1. How do I do this?

Also does the dot in the middle mean multiply? It's about this height: - and its a dot, like m = k . n, only the dot is higher. I've never seen this notation before but I am currently assuming it does mean multiply. What is the significance, and why can't you just wrote m = kn?
2. When it says rearrange and find 2 numbers that multiply to make 1, it basically means solve the quadratic by factorisation.
In the set of integers the only numbers that multiply to make 1 are
1*1 = 1
(-1)*(-1) = 1
This means that there are only 2 possible factorisations;
(x-1)(x-1) or (x+1)(x+1). From this, you can find the possible k's

Yes, the dot means multiply.
3. ok thanks
4. (Original post by JamesF)
When it says rearrange and find 2 numbers that multiply to make 1, it basically means solve the quadratic by factorisation.
In the set of integers the only numbers that multiply to make 1 are
1*1 = 1
(-1)*(-1) = 1
This means that there are only 2 possible factorisations;
(x-1)(x-1) or (x+1)(x+1). From this, you can find the possible k's

Yes, the dot means multiply.
where did you get those two factorisations from? neither multiply out to make x^2 + 1, and if the equation is x^2 + 1 = kx why do you care what two integers multiply to make 1?
5. x^2 + 1 = kx
0 = x^2 - kx + 1
6. so you're factorising x^2 - kx + 1 , and that only 1*1 or -1*-1 = 1 tells you that x^2 - kx + 1 = (x+1)(x+1) or (x-1)(x-1), so therefore

x^2 - kx + 1 = x^2 + 2x + 1
- k = 2
k = -2

or

x^2 - kx + 1 = x^2 - 2x + 1
- k = -2
k = 2?

It's making more sense now, but I don't understand how you can equate x^2 - kx + 1 to the two factorisations.

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