Question

prove that every positive integer different from 1 can be expressed as a product of a non negative power of 2 and an odd number.

Answer 1

case 1

if the given number is a factor of 2 then

$we \: can \: write \: it \: as \: {2}^{n} \times a \: where \: n \: is \: positive \: and \: a \: is \: a \: odd \: no$

case 2



if given no is not a factor of 2 I.e. it is odd no then
$we \: can \: write \: the \: no \: as \: {2}^{0} \times a \: where \: a \: is \: a \: odd \: no \: which \: is \: equal \: to \: a$

hence proved

Answer 2

Answer:

If we now write n as k. 2^m, then k has to be an odd integer because if k is even, then at least one power of 2 will divide k and so 2^(m+1) will divide n contradicting our choice of m as the highest power of 2 dividing n. Thus n can be uniquely written as a non-negative power of 2 multiplied by an odd integer.