Math, asked by beastboypush2211, 8 months ago

Prove that 15 is irrational​

Answers

Answered by vk5219167
0

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

Answered by mangalasingh00978
4

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.

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