My teacher recently (a year ago) gave me some sheets with problems about coprime numbers, however, without help I couldn't solve even one problem. It is quite sad because I usually don't feel so beat down by math that I wouldn't even get an idea of where to start with solving a problem. That got me thinking that perhaps I lack understanding of coprimes. Perhaps anyone know great resources for learning some stuff about them? Keep in mind that I am not really advanced in math and that all things about coprimes that I know are that greatest common divisor of all the coprime numbers are 1 (recently got into phi function as well, but that is not helping me at all in doing those problems :( ).
Examples of problems that I am struggling with:
1: Prove that if \(a\) and \(b\) are coprime, then \(a^m\) and \(b^m\) are also coprime, if \(m\) is any natural number.
2: Prove that if \(a\) is a prime number, which isn't equal to 2 nor 3, then \(a^2 - 1\) is a multiple of 24. (Oh, I think that I lack of understanding of prime numbers as well)
And quite a lot more. (I understand when someone give the solution to problem, but I can't grasp any of them on my own...)
If you don't know any books, but can give me some advice, I would be glad to read it :)