Prove that 15 is irrational
Answers
Answer:
This proof uses the unique prime factorisation theorem that every positive integer has a unique factorisation as a product of positive prime numbers. Suppose √15=pq for some p,q∈N . ... Now k<q<p contradicting our assertion that p,q is the smallest pair of values such that √15=pq
Answer:
This proof uses the unique prime factorisation theorem that every positive integer has a unique factorisation as a product of positive prime numbers.
Suppose
√
15
=
p
q
for some
p
,
q
∈
N
. and that
p
and
q
are the smallest such positive integers.
Then
p
2
=
15
q
2
The right hand side has factors of
3
and
5
, so
p
2
must be divisible by
3
and by
5
. By the unique prime factorisation theorem,
p
must also be divisible by
3
and
5
.
So
p
=
3
⋅
5
⋅
k
=
15
k
for some
k
∈
N
.
Then we have:
15
q
2
=
p
2
=
(
15
k
)
2
=
15
⋅
(
15
k
2
)
Divide both ends by
15
to find:
q
2
=
15
k
2
So
15
=
q
2
k
2
and
√
15
=
q
k
Now
k
<
q
<
p
contradicting our assertion that
p
,
q
is the smallest pair of values such that
√
15
=
p
q
.
So our initial assertion was false and there is no such pair of integers.