Question

Prove that every positive integer different from 1 can be expressed as a product of non negative power of 2 and an odd number .

Answer 1

let n be a positive integer different from 1 

then it can be expressed as : n= p1a1p2a2p3a3......pkak

there can be 2 different possibilities

1)-    when p1 = 2 and  p1a1p2a2p3a3......pkak are odd positive integers

        n= 2a1 x  ( p1a1p2a2p3a3...... pkak )

        n = ( a non- negative power of 2) x (an odd positive integer)

2)-   whem each of p1a1p2a2p3a3......pkak are odd positive integers

       n = p1a1p2a2p3a3......pkak

       n = 20 x p1a1p2a2p3a3......pkak

       n =  ( a non- negative power of 2) x (an odd positive integer)

Answer 2

Any odd number can be written as the product of a non­negative power of 2 and an odd number. Let n be an odd number. Then n =
 2↑0xn=1*n=n
For even numbers, we can follow the Fundamental Theorem of Arithmetic. The theorem states that every integer can be written as the product of prime numbers. 
It says that an even number n will have 2 as a prime factor. Let n be an even number where n is a power of two and can be be written as n =
2*k*1
, since 1 is an odd number, and the prime factorization of n =
2k*p1*p2.................pn
where k is some positive integer, and p1…………, pn are primes. Since p1, ..., pn are prime numbers greater than two, they are odd. Therefore their product will also be odd. Thus n can be written as the product of a power of two and an odd number.