Three proofs about primes and threes

qtpipi · #1 $Old$ April 9th, 2009, 07:45 PM

I thought this class would be fun but it turned out to be not the thing for me. It's too late to drop the course so I'm stuck doing homework that I don't understand.

Any help with any of these would be appreciated.

Grandad · #2 $Old$ April 9th, 2009, 11:42 PM

Hello qtpipi

Quote:

Originally Posted by qtpipi $View Post$

I thought this class would be fun but it turned out to be not the thing for me. It's too late to drop the course so I'm stuck doing homework that I don't understand.

Any help with any of these would be appreciated.

(a) Suppose that $n$ is represented by the sequence of digits $a_na_{n-1}...a_1$ , and that $s = a_n + a_{n-1}+...+a_1$

Then $n = 10^{n-1}a_n + 10^{n-2}a_{n-1} + ... + a_1$

$\Rightarrow n-s = (10^{n-1}-1)a_n +(10^{n-2}-1)a_{n-1} + ... +(1-1)a_1$

Now $10 = 1 \mod 9 \Rightarrow 10^i = 1 \mod 9,\forall i \in \mathbb{N}$

$\Rightarrow 10^i-1 = 0 \mod 9$

$\Rightarrow n-s = 0 \mod 9$

$\Rightarrow n$ and $s$ leave the same remainder when divided by $9$

$\Rightarrow 3|n \iff 3|s$ (and of course, $9|n \iff 9|s$ as well).

(b) In any three consecutive integers, $(x-1), x, (x+1)$ , one is always a multiple of $3$ . And $(x-1)$ and $(x+1)$ are primes, and $(x-1) \ne 3 \Rightarrow$ neither $(x-1)$ nor $(x+1)$ is a multiple of $3$

$\Rightarrow 3|x$

And, of course, $x$ is even whenever $(x-1)$ is a prime $> 2$ .

Hence $6|x$

(c) Consider the remainders when (x-1) and (x+1) are divided by 3, and then use the results of (a).

Grandad