Prove that every positive integers different …

Sol: Let n be a positive integer then, n can be odd or even. (i) If n is odd, it is not divisible by 2.   Therefore, n can be written as = 1 x n = 20 x n = product of a non-negative power of 2 and an odd number .   (ii) If n is even, it is divisible by 2.   Then m = n / 2 is an integer.    If m is odd, it cannot be divided by 2.   Because of m = n / 2 ⇒ n = 2m = 21 x m = product of a non-negative power of 2 and an odd number.  If m is even, it is divisible by 2.   Then p = m / 2 is an integer.   If p is odd, it cannot be divided by 2.   Because p = m / 2 and m = n / 2, we obtain p = n / 4 ⇒ n = 4p = 22 x p = product of a non-negative power of 2 and an odd number   If p is even, it is further divisible by 2, and the above steps can be repeated until we arrive at an integer which is no longer divisible by 2, i.e., it is odd.      Hence every positive integer different from 1 can be expressed as a product of non-negative power of 2 and an odd number.