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

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.

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

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.

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