Number theory & modular arithmetic worksheet with answers

Free printable Number theory & modular arithmetic worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy

Find (−22) mod 3, the remainder r with 0 ≤ r < 3.

a = −22n = 3

(−22) mod 3 =

2.Easy

Find 897 mod 9, the remainder r with 0 ≤ r < 9.

a = 897n = 9

897 mod 9 =

3.Medium

Find the inverse of 49 modulo 89: the number x with 0 < x < 89 and 49·x ≡ 1 (mod 89).

a = 49n = 89

49⁻¹ mod 89 =

4.Medium

Find the smallest positive integer x with x ≡ 3 (mod 5) and x ≡ 2 (mod 9).

x ≡ 3 (mod 5)x ≡ 2 (mod 9)

x =

5.Hard

59 is prime. Use Fermat’s little theorem to find 19⁴⁵⁰⁷⁹ mod 59.

p = 59k = 45079

19⁴⁵⁰⁷⁹ mod 59 =

6.Hard

Encrypt the message m = 508 with the RSA public key (n, e) = (1081, 13). Find the ciphertext c.

n = 1081e = 13m = 508

c =

Answer key

1. (−22) mod 3 = 2
2. 897 mod 9 = 6
3. 49⁻¹ mod 89 = 20
4. x = 38
5. 19⁴⁵⁰⁷⁹ mod 59 = 57
6. c = 349

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