Elementary Number Theory – Tutorial Exercises
1. Evaluate 6*7 mod 11.
2. Evaluate 5*8 mod 17.
3. Evaluate 6/7 mod 11.
4. Evaluate ¾ mod 17.
5. Evaluate ¾ mod 19.
6. Evaluate 6/9 mod 13.
7. Calculate 1033 mod 13.
8. Evaluate 7643514 mod 10.
9. What is the least-significant decimal digit of 10023755?
10. Evaluate 877226001 mod 12.
11. Evaluate 10154 mod 103.
12. Evaluate 3243100003 mod 8.
13. Evaluate 10166 mod 127.
14. Evaluate 87672123 mod 15.
15. What is 101108 mod 109? (Note 109 is a prime).
16. Evaluate a square root of 3 mod 83.
17. Evaluate the cube root of 2 mod 83.
18. Find all the square roots of 3 mod 143.
19. Evaluate a square root of 3 mod 59.
20. Evaluate the cube root of 2 mod 59.
21. Find the cube root of 18 mod 55.
22. Outline the steps in the Euclidean GCD algorithm for the
calculation of GCD(40902,24140).
23. Find x such that Ord43(x)=7.
24. Find x such that Ord29(x)=7.
25. What is Φ(28), where Φ is the Euler Totient function?
26. Find all the square roots of 36 mod 143.
27. Find all the square roots of 11 mod 35.
28. Find a primitive root mod 19.
29. List the quadratic residues mod 23.
30. Is 7 a quadratic residue mod 19 ?
31. Is 2 a quadratic residue mod 109?
32. Is 2 a quadratic residue mod 19? Explain your answer.
33. Is 2 a quadratic residue mod 91? Explain your answer.
34. A number x less than 91 has a remainder of 3 when divided by 7
and a remainder of 4 when divided by 13. What are the possible
values of x? Use the Chinese Remainder Theorem.
35. 3x mod 19 = 13, find the smallest x which satisfies this.
36. Euler tells us that the jacobi symbol (2/n) = (-1)(n*n-1)/8. Use this fact
to determine whether or not 2 is a quadratic residue with respect to
a prime modulus p, for each of the cases
•
•
•
•
p ≡ 1 mod 8
p ≡ 3 mod 8
p ≡ 5 mod 8
p ≡ 7 mod 8
37. Under what condition on p will a number have a unique cube root
mod p, where p is a prime number? Derive an explicit formula for
finding cube roots when this condition is met.
38. Working with the prime modulus 463, find a generator of the primeorder subgroup of size 11.
39. Find the two square roots of 38 mod 103. Note that 103 is a "3 mod
4" prime.
40. If n = p.q where p and q are primes, derive an expression for φ (n),
the Euler Totient function.