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 (−80) mod 7, the remainder r with 0 ≤ r < 7.

a = −80n = 7

(−80) mod 7 =

2.Easy

Use Euclid’s algorithm to find gcd(318, 294).

a = 318b = 294

gcd(318, 294) =

3.Medium

Find the inverse of 7 modulo 39: the number x with 0 < x < 39 and 7·x ≡ 1 (mod 39).

a = 7n = 39

7⁻¹ mod 39 =

4.Medium

Find the inverse of 42 modulo 47: the number x with 0 < x < 47 and 42·x ≡ 1 (mod 47).

a = 42n = 47

42⁻¹ mod 47 =

5.Hard

19 is prime. Use Fermat’s little theorem to find 18⁴⁷¹⁶⁹ mod 19.

p = 19k = 47169

18⁴⁷¹⁶⁹ mod 19 =

6.Hard

How many solutions N does 18x ≡ 10 (mod 50) have in {0, 1, …, 49}?

18x ≡ 10 (mod 50)

N =

Answer key

1. (−80) mod 7 = 4
2. gcd(318, 294) = 6
3. 7⁻¹ mod 39 = 28
4. 42⁻¹ mod 47 = 28
5. 18⁴⁷¹⁶⁹ mod 19 = 18
6. N = 2

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