# Math Help - Direct Proof

1. ## Direct Proof

Hi people,

I think its true that for every positive natural number, lets call it n, can be put in form of 2^k * m (where m is an odd number).

I think thats correct? Please confirm...

How would you use direct proof to proof this?? I cant get grips of it. Its abit advanced for me

Thanks.

2. If we consider $1$ an odd number your are right...

Kind regards

$\chi$ $\sigma$

3. if n is odd choose k = 0 and m = n.
if n is even then write $n$ in its prime decomposition.

$n=2^rp_1p_2p_3\dots$

Choose k=r and choose $m$ to be the product of primes, $p_i$. The product of primes is guaranteed to be odd because an odd times an odd is an odd and no primes other than 2 are even. If there are no $p_i$s (ie. n is a power of 2), then take m=1.