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

Use Euclid’s algorithm to find gcd(672, 180).

a = 672b = 180

gcd(672, 180) =

2.Easy

Find the least common multiple lcm(552, 120). Use Euclid’s algorithm for the gcd first.

a = 552b = 120

lcm(552, 120) =

3.Medium

Solve 13x ≡ 5 (mod 31). Give the solution x with 0 < x < 31.

13x ≡ 5 (mod 31)

x =

4.Medium

Solve 22x ≡ 9 (mod 47). Give the solution x with 0 < x < 47.

22x ≡ 9 (mod 47)

x =

5.Hard

71 is prime. Use Fermat’s little theorem to find 9³⁵²³³ mod 71.

p = 71k = 35233

9³⁵²³³ mod 71 =

6.Hard

Use repeated squaring to find 19¹²⁵ mod 63.

a = 19k = 125n = 63

19¹²⁵ mod 63 =

Answer key

1. gcd(672, 180) = 12
2. lcm(552, 120) = 2760
3. x = 29
4. x = 41
5. 9³⁵²³³ mod 71 = 2
6. 19¹²⁵ mod 63 = 10

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