Any positive integer can be represented uniquely as a product with such that 2 does not divide . In other words, is the "odd" part of and is the "even" part of . Here (Link) is information on the "Odd Part" of a number. Here (Link) is information on the "Even Part" of a number. It is obvious that for any choice of nonnegative integers such that is odd. Next, let with odd positive integers and nonnegative integers. Since must divide and 2 does not divide , it must be that . Similarly, since must divide and 2 does not divide , it must be that . Hence, . Then, by cancellation, . This shows that is one-to-one. To show that is onto, let be any natural number. Let be the highest power of that divides . Since is a natural number, . Let . Now, . Since is an odd positive integer, is an even positive integer, so is a positive integer. Since is a nonnegative integer, is a positive integer. Hence, , so is onto.