Theory NO. 1 : Number Theory

19Feb08

Prove that for every natural number n \geq there exists n different natural numbers such as sum of any two is divisable by their diffrence.

Proof:
Because number a+b is divisable by a - b , we can write a+b \equiv 0 \mod{a-b} and of course a-b \equiv 0 \mod{a-b}. Adding those two together:
a+b + a - b = 2a \equiv a - b \equiv \ 0 \mod{a-b} \Leftrightarrow 2a \equiv 0 \mod{a-b}

Induction:
1. Sequence 1,2 works
2. Assume that sequence x_{1}, x_{2}, ..., x_{n} works, then the number 2x_{1}x_{2}...x_{n} is divisable by the difference of any two numbers from the sequence
3. Let p=x_{1}x_{2}...x_{n}, then a sequence, including n+1 numbers:
p, p + x_{1}, p + x_{2}, ..., p + x_{n}
‘Works’.



One Response to “Theory NO. 1 : Number Theory”

  1. 1 Nadya Fermega

    I have a problem, how to you prove that 1 – 1 = 0 without using the above number theory, can you help me?


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