Prove that every positive integer is uniquely representable as the product of a nonnegative power of 2(perhaps 2^0) and an odd number.
Let n be a positive integer.
Then, n can be odd or even.
(i) If n is odd, it is not divisible b y 2.
We can write n = 1*n = 20*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 m = n/2, n = 2m = 21*m
= product of a non-negative power of 2 and an odd number
If m is even, it is divisible b y 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 get p = n/4, or n = 4p = 22*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 proved that every positive integer is uniquely representable as the product of a
non-negative power of 2 and an odd number.