Nothing is well understood unless we can apply it to solve some problems. Currently, I am learning number theory and there is no good classified list of problems for number theory. So, It is difficult to find some problem based on a specific topic.

Here I will maintain a list of problems of number theory in a few category. So, In near future, nobody have to face the same problem. Contribute interesting problems, people(mostly me :D ) will be grateful to you.

1. Divisibility/Primes/Co-prime/GCD/LCM
Problems:
1. LOJ: 1014, 1035
2. CF: 230B, 711E, 585E, 585C, 222C, 546D, GYM101212B
3. SPOJ: LCMSUMH
4. E-OLYMP: 1146, 1243, 1147, 1229, 1244, 1230

2. Continued Fraction
Problems:
1. UVA: 834
2. HackerRank: euler064

3. Diophantine Equations
Problems:
1. LOJ: 1077
2. CF: 7C

4. Functions: Counting Divisors/Divisor Sum
Problems:
1. LOJ: 1054
2. UVA: 13083

5. Function: Totient and Similar
Problems:
1. LOJ: 1007
2. SPOJ: NFACTOR , NDIV

6. Modular Arithmetic/CRT
Problems:
1. LOJ: 1067
2. CF: 687B, GYM100825A

7. Discrete Logarithms
Problems:
1. LOJ: 1325

8. Number Systems
Problems:
1. LOJ: 1028
2. E-OLYMP: 1001, 1002, 1008, 1013

9. Primitive Root
Problems:
1. E-OLYMP: 647

NB: This list is just created. And hopefully will grow with user contribution and as I learn. Thanks for being a part of it. :)

Contributors:
HaabyHings, Rudy1444, Lucas97, Batman, t3rminated, MladenP, TooNewbie