Hello, can anyone help me with these two problems?? Thank you so much in advance.

1) Prove: If x ≡ y (mod m), then (x, m) = (y, m)

2) Show that if n > 4 is not prime, then (n-1)! ≡ 0 (mod n).

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- January 22nd 2007, 04:36 PMjenjencongruence modulo m
Hello, can anyone help me with these two problems?? Thank you so much in advance.

1) Prove: If x ≡ y (mod m), then (x, m) = (y, m)

2) Show that if n > 4 is not prime, then (n-1)! ≡ 0 (mod n). - January 22nd 2007, 04:45 PMThePerfectHacker
Let .

I presume it means .

We know that,

thus, for some .

Let .

Let .

We will prove it by trichtonomy.

Assume .

But that cannot be because,

And and .

Thus, thus . And we also know that .

Thus, because . And hence it is not "greatest".

By similar reasoning. We can show leads to contradiction.

Thus, by trichtonomy,

- January 22nd 2007, 05:18 PMThePerfectHacker
There are two possibilities.

1)n is not a square.

2)n is a square.

If n is not a square then it has a non-trivial proper factorization where .

Where are distinct.

Thus among the factors of :

We can find its factors .

When is a square there are 2 possibilities.

1)n is not a square of a prime.

2)n is a square of a prime.

If n is not a square of a prime we know that where because it is not a prime. Thus, it has a factorization in the form where are distinct and the same argument applies.

If n is a square of a prime then we have a minor problem. For example if n=4=2^2 it does not work. Which is why the initial conditions of the problem says n>4. Thus we know that and in no other way. We know that contains one factor. But what about other another factor of ? It turns out that when the factor appears among . Thus, there is another number that has a factor of . The reason why is because for thus, and is hence among one of the factors. - January 22nd 2007, 07:31 PMjenjen
thanks theperfecthacker for the quick reply!

- February 19th 2007, 09:51 PMIdeasman
Hmm, you have:

2 <= a,b <= n - 1, shouldn't it be: 1 < a < b < n - 1? Maybe you got it mixed up for when it is a square. The proof I was given was:

**Either n is a perfect square, n = a^2 in which case 2 < a < 2a <= n−1 and hence a and 2a are among the numbers {3,4, . . . ,n−1} or n is not a perfect square, but still composite, with n = ab, 1 < a < b < n−1.**

Does this work? It contradicts a few of your results. - February 20th 2007, 11:37 AMThePerfectHacker
- February 21st 2007, 09:22 PMIdeasman
Sorry to be a pain, but I clearly understand your proof now TPF. What I don't understand is how the proof I was given (below) proves the claim; how does that tie it all together showing that n is composite if n > 4 and that n will then divide (n - 1)!

Proof:

Either n is a perfect square, n = a^2 in which case 2 < a < 2a <= n−1 and hence a and 2a are among the numbers {3,4, . . . ,n−1} or n is not a perfect square, but still composite, with n = ab, 1 < a < b < n−1. - February 22nd 2007, 03:40 PMThePerfectHacker
- February 22nd 2007, 03:57 PMIdeasman