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 the least common multiple lcm(88, 38). Use Euclid’s algorithm for the gcd first.

a = 88b = 38

lcm(88, 38) =

2.Easy

Use Euclid’s algorithm to find gcd(609, 441).

a = 609b = 441

gcd(609, 441) =

3.Medium

Find the inverse of 4 modulo 11: the number x with 0 < x < 11 and 4·x ≡ 1 (mod 11).

a = 4n = 11

4⁻¹ mod 11 =

4.Medium

Find the inverse of 6 modulo 17: the number x with 0 < x < 17 and 6·x ≡ 1 (mod 17).

a = 6n = 17

6⁻¹ mod 17 =

5.Hard

How many solutions N does 22x ≡ 30 (mod 50) have in {0, 1, …, 49}?

22x ≡ 30 (mod 50)

N =

6.Hard

Solve 16x ≡ 12 (mod 18). Give the smallest positive solution x.

16x ≡ 12 (mod 18)

x =

Answer key

1. lcm(88, 38) = 1672
2. gcd(609, 441) = 21
3. 4⁻¹ mod 11 = 3
4. 6⁻¹ mod 17 = 3
5. N = 2
6. x = 3

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