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# What does this analysis even mean prime numbers/rational numbers

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Page 4 in the book.

It says. "Then there exists a prime number p which divides n an odd number of times" in the context what does that even mean :l
2. If n is not a perfect square, and every prime number divides into n an even number of times, then n is actually a perfect square. So there has to exist at least one prime number which divides into n an odd number of times.
3. Everything is easier when you try out some examples.

Pick a square number: 144
What's its prime factorisaton: 144 = 12x12 = (2x6)*(2x6) = 2x(2x3)x2x(2x3) = (2^4)*(3^2)

That tells you that the prime number 2 divides 144 four times, i.e. 144/2 = 72, 72/2 = 36, 36/2 = 18, 18/2 = 9.
It also tells you the prime number 3 divides 144 two times, i.e. 144/3 = 48, 48/3 = 16.

four and two are even numbers, which is what the book says (even numbers of prime factors = square).

Now pick a non-square number: 200
The prime factorisation (same process) is: 200 = 2*2*2*5*5

So the prime number 2 divides 200 three times! Which is an odd number of times. That means 200 cannot be a square number.

Why? Because, to be a perfect square, it should be possible to write the prime factorisation as (something)^2. With 144, you can write it as (2x2x3)^2, but you cannot do this with 200.
4. Thank you

(Original post by mik1a)
Everything is easier when you try out some examples.

Pick a square number: 144
What's its prime factorisaton: 144 = 12x12 = (2x6)*(2x6) = 2x(2x3)x2x(2x3) = (2^4)*(3^2)

That tells you that the prime number 2 divides 144 four times, i.e. 144/2 = 72, 72/2 = 36, 36/2 = 18, 18/2 = 9.
It also tells you the prime number 3 divides 144 two times, i.e. 144/3 = 48, 48/3 = 16.

four and two are even numbers, which is what the book says (even numbers of prime factors = square).

Now pick a non-square number: 200
The prime factorisation (same process) is: 200 = 2*2*2*5*5

So the prime number 2 divides 200 three times! Which is an odd number of times. That means 200 cannot be a square number.

Why? Because, to be a perfect square, it should be possible to write the prime factorisation as (something)^2. With 144, you can write it as (2x2x3)^2, but you cannot do this with 200.
(Original post by Alex:)
If n is not a perfect square, and every prime number divides into n an even number of times, then n is actually a perfect square. So there has to exist at least one prime number which divides into n an odd number of times.

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Updated: September 24, 2016
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