Question

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

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Solution

## Let n be positive integer other then 1. By the fundamental theorem of Arithmetic n can be uniquely expressed as power of primes gin ascending order so, letn=p1a1p2a2...pkak can be the unique factorisation of n into primes with p1<p2<p3...<pk.Clearly, either p1=2 and p2,p3...<pk are odd positive integers or each of p1,p2...,pk is an odd positive integer.Therefore, we have the following cases:Case (I) When p1=2 and p2,p3...,pk are odd positive integers: In this case, we haven=2a1p2a2p3a3...pkak⇒ n=2a1×(p2a2p3a3...pkak) An odd positive integer.⇒ n=2a1× An odd positive integer.⇒ n= (A non-negative power of 2) × (An odd positive integer)Case (II) When each of p1,p2,p3,...,pk is an odd positive integer:In this case, we haven=p1a1p2a2p3a3...pkak ⇒ n=20×(p1a1p2a2p3a3...pkak)⇒ n= (A non-negative power of 2) × (An odd positive integer)Hence, in either case n is expressible as the product of a non-negative power of 2 and positive integer.

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