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Help: finding materials to study coprime numbers

Hello, everybody.

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 :)

Note by Zyberg Nee
1 year, 7 months ago

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First example: As \(\gcd(a,b)=1\), a and b have no common prime factors. They won't have any common prime factors when they are raised to a natural number. The result follows.

Second example: All prime numbers>3 are of the form \(6n \pm 1\) where n is a positive integer.

So, \(a^2-1=36n^2 \pm 12n=12n(3n \pm 1)\).

If n is even, 24 all divide 12n and if n is odd, 24 will divide \(12(3n \pm1)\).

Proved.

Brilliant Member - 1 year, 7 months ago

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Check out greatest common divisor and euclidean algorithm.

Calvin Lin Staff - 1 year, 7 months ago

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Thank you very much!

Zyberg Nee - 1 year, 7 months ago

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