matt8 wrote:
Bunuel wrote:
vikram4689 wrote:
Both are correct examples but the one i gave does not follow what you mentioned
You said since 2*P is divisible by 5, therefore P is divisible by 5... This is not correct P=5/2
Not so. Your example is NOT correct.
I'm saying that if p is an integer and 2p is divisible by 5 then p must be divisible by 5. Now, check your example, is it correct?
It should be: 2*5 is divisible by 5 (p=5) so 5 (p) is divisible by 5.
Hope it's clear.
I believe vikram4689 is correct. If we know that 2P is divisible by 5, then we only know that P is divisible by 2.5.
Let,
P=x^2+y^2 and
N=5a^2+2a+5b^2+4b+1
Therefore, we have already proven:
2P=5N
P=(5/2)N
The only way for P to be divisible by 5 is if N is even (contains a factor of 2). Unfortunately, we can't factor out a 2 from N. Therefore, whether or not N is even depends on the values of A and B.
By looking at the equation of N, we can see that it will be even only if A and B are not both positive or both negative.
It works out that the equations (x-y)=5a+1 and (x+y)=5b+2 can both be satisfied only when A and B are not both positive or both negative. This is why Bunuel's incorrect assumption still works in this case.
There is no way I could have figured all of this out in 2 or 3 minutes though.
Think again - It's not 'Bunuel's incorrect assumption'. It is given that x and y are integers so, given that P=x^2+y^2,
P will be a positive integer.
Now that we know that P is an integer, if 2P is divisible by 5, P must be divisible by 5.
Given that a and b are positive integers and that 'a' is divisible by 'b', a MUST have b as a factor. It's a basic mathematical fact. When I say, 'a' is divisible by 'b', I mean 'a' is a multiple of 'b' which means 'b' is a factor of 'a'. Try some examples to convince yourself.
This statement has far reaching implications.
Think of questions like: Is \(2^{100}\) divisible by 3? I will say 'no' without a thought.
Is \(2^{10}*3^{100}*5\) divisible by 7? Again, absolutely no!
To reiterate, if 'a' is to be divisible by 'b', a must have b as a factor. (talking about positive integers)