# show that every positive integer is uniquely representable as the product of a non-ne

• Sep 30th 2013, 05:45 PM
virebbala90
show that every positive integer is uniquely representable as the product of a non-ne
show that every positive integer is uniquely representable as the product of a non-negative power of two and an odd integer.

we are looking for a direct proof which does not use the Fundamental Theorem of Algebra (i.e. the uniqueness of prime factorizations). better to use a mathematical induction method. please its urgent
• Sep 30th 2013, 06:03 PM
HallsofIvy
Re: show that every positive integer is uniquely representable as the product of a no
The obvious thing to do is repeatedly divide by 2 until the remainder is odd.

To do that you will need 'Let N be a positive integer. Then there exist a unique non-negative integer k such that \$\displaystyle 2^k\le N< 2^{k+1}\$". That can be proved by induction on N.
• Oct 1st 2013, 07:34 AM
virebbala90
Re: show that every positive integer is uniquely representable as the product of a no
Thanks for your help. can you please elaborate te answer. i have short time to submit homework
• Oct 1st 2013, 07:41 AM
emakarov
Re: show that every positive integer is uniquely representable as the product of a no
The easiest is to use strong induction. Assume the claim is proved for all k < n. If n is odd, the claim is proved. Otherwise, n = 2k where k < n. Apply the induction hypothesis to k and derive from it the claim for n.
• Oct 1st 2013, 11:23 PM
topsquark
Re: show that every positive integer is uniquely representable as the product of a no
Quote:

Originally Posted by virebbala90
i have short time to submit homework

We are not here to do your homework. If you show some effort we can give you pointers, but no more.

-Dan