Prove that every positive integer different from 1 can be expressed as a product
a non-negative power of 2 and an odd number.
Answers
Answer:
ANSWER
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, let
n=p
1
a
1
p
2
a
2
...p
k
a
k
can be the unique factorisation of n into primes with p
1
<p
2
<p
3
...<p
k
.
Clearly, either p
1
=2 and p
2
,p
3
...<p
k
are odd positive integers or each of p
1
,p
2
...,p
k
is an odd positive integer.
Therefore, we have the following cases:
Case (I) When p
1
=2 and p
2
,p
3
...,p
k
are odd positive integers: In this case, we have
n=2
a
1
p
2
a
2
p
3
a
3
...p
k
a
k
⇒ n=2
a
1
×(p
2
a
2
p
3
a
3
...p
k
a
k
) An odd positive integer.
⇒ n=2
a
1
× An odd positive integer.
⇒ n= (A non-negative power of 2) × (An odd positive integer)
Case (II) When each of p
1
,p
2
,p
3
,...,p
k
is an odd positive integer:
In this case, we have
n=p
1
a
1
p
2
a
2
p
3
a
3
...p
k
a
k
⇒ n=2
0
×(p
1
a
1
p
2
a
2
p
3
a
3
...p
k
a
k
)
⇒ 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.
Step-by-step explanation:
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