Sample questions about GCD, Euclid's algorithm, Moduli, The fundamental theorem of arithmetic

Question

I need many questions and exercises about GCD, Euclid's algorithm, Moduli, The fundamental theorem of arithmetic, etc., for my test preparation.

An example question:

Prove that $(n + 1)! + 1$ and $n! + 1$ are relatively prime, i.e. : $$\gcd[(n + 1)! + 1, n! + 1] = 1$$

Can you refer me to text books, online courses, past exams, and any other source where I can find such practice questions?

Ideally there will be solutions as well.

Thanks!

Answer 1

Burton's Elementary Number Theory has a bunch of beginner-friendly exam-type practice problems and the solutions manual can be found online.

Answer 2

Two books I'd recommend are Elementary Number Theory and Its Applications by Kennet H. Rosen, with mostly beginner problems, and 1001 Problems in Classical Number Theory by Armel Mercier and Jean-Marie De Koninck, with a very wide variety of problems. Both of which can be found online

Answer 3

I M Vinogradov, Elements of Number Theory. "One of the most valuable characteristics of this book is its stress on learning number theory by means of demonstrations and problems. More than 200 problems and full solutions appear in the text, plus 100 numerical exercises." Some of the exercises are quite difficult.

Another book with a similar structure, and not as demanding, is Joe Roberts, Elementary Number Theory: A Problem Oriented Approach, https://mitpress.mit.edu/9780262680288/elementary-number-theory/