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 ". That can be proved by induction on N.
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
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 ". That can be proved by induction on N.
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.