Use the Fundamental Theorem of Arithmetic to show every positive integer, n>0, can be written uniquely as the product of a power of 2 and an odd number.
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.
Oh, I guess I just overlooked that part because it seemed obvious. Write a number as its unique product of primes:
Now, using associativity, this is equal to
The left term is a power of two and the right term is an odd number, because every product of only odd numbers is odd. You can prove that statement easily using modular arithmetic.