Number theory factorization proof

[dancergirlie]

Homework Statement

1. Homework Statement

If n is a nonzero integer, prove that n cam be written uniquely in the form n=(2^k)m, where It is in the primes and unique factorization chapter so maybe that every integer n (except 0 and 1) can be written as a product of primes

Homework Equations

It is in the primes and unique factorization chapter so maybe that every integer n (except 0 and 1) can be written as a product of primes

The Attempt at a Solution

I am not sure, but I tried to do a proof by induction on n

Let n=1
so, 1=(2^k)(m)
1=(1)(1)=1 where k=0 and m=1
so the statement holds for n=1

Now assume the statement holds for n=(r-1) for any integer (except 0 and 1)
so
(r-1)=(2^k)(m) where k is greater than or equal to zero and m is odd

Now let n=r
so r=(2^k)(m)

This is where I don't know what to do... that is why I'm thinking there is an easier way to do this besides induction. Any help would be appreciated =)

 

[Mark44]

You left off the conditions in your problem statement.

If n is a nonzero integer, prove that n cam be written uniquely in the form n=(2^k)m, where

After this you tell us where the problem appeared in your text. What are the conditions on k and m?

Also, for your induction proof, I would start with 2, not 1.

 

[dancergirlie]

ohh I'm sorry

k has to be greater than or equal to zero and m is odd

 

[dancergirlie]

Let me repost the problem, cause I see some typos...

Let n be a nonzero integer, prove that n can be written uniquely in the form (2^k)(m) for any integer k greater than or equal to zero and for any odd integer m.

If i can avoid doing an induction proof, that would be great, but if this is the only way then i guess i can do it

 

[Mark44]

This might not pass muster as a formal proof, but it will help you think about this problem.

There are two cases: n is even and n is odd.
1) If n is odd, it has no factors of 2, so n = 2⁰*n, and you're done.
2) If n is even, there must be at least one factor of 2, so n = 2¹*m
a) If m is odd, you're done.
b) If m is even, apply step 1 again.