Suppose that there are two distinct representations of the same number as the product of a power of 2 and an odd number (let us call them and . This number has a unique decomposition into primes, from which it immediately follows that , since and contain no factors of 2, and 2 is prime. Thus . Now , and since , it follows that , thus demonstrating that the two representations are not distinct, and hence the number has a unique representation as a power of 2 and an odd number.