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(741, 559). Use Euclid’s algorithm for the gcd first.

a = 741b = 559

lcm(741, 559) =

2.Easy

Find the least common multiple lcm(147, 91). Use Euclid’s algorithm for the gcd first.

a = 147b = 91

lcm(147, 91) =

3.Medium

Find the inverse of 8 modulo 29: the number x with 0 < x < 29 and 8·x ≡ 1 (mod 29).

a = 8n = 29

8⁻¹ mod 29 =

4.Medium

Solve 27x ≡ 14 (mod 73). Give the solution x with 0 < x < 73.

27x ≡ 14 (mod 73)

x =

5.Hard

53 is prime. Use Fermat’s little theorem to find 5⁹⁷¹²⁰ mod 53.

p = 53k = 97120

5⁹⁷¹²⁰ mod 53 =

6.Hard

How many solutions N does 24x ≡ 12 (mod 42) have in {0, 1, …, 41}?

24x ≡ 12 (mod 42)

N =

Answer key

1. lcm(741, 559) = 31863
2. lcm(147, 91) = 1911
3. 8⁻¹ mod 29 = 11
4. x = 60
5. 5⁹⁷¹²⁰ mod 53 = 49
6. N = 6

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