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(767, 91). Use Euclid’s algorithm for the gcd first.

a = 767b = 91

lcm(767, 91) =

2.Easy

Use Euclid’s algorithm to find gcd(62, 44).

a = 62b = 44

gcd(62, 44) =

3.Medium

Solve 24x ≡ 26 (mod 47). Give the solution x with 0 < x < 47.

24x ≡ 26 (mod 47)

x =

4.Medium

Find the inverse of 33 modulo 71: the number x with 0 < x < 71 and 33·x ≡ 1 (mod 71).

a = 33n = 71

33⁻¹ mod 71 =

5.Hard

Use repeated squaring to find 66¹⁷⁴ mod 73.

a = 66k = 174n = 73

66¹⁷⁴ mod 73 =

6.Hard

Solve 30x ≡ 27 (mod 63). Give the smallest positive solution x.

30x ≡ 27 (mod 63)

x =

Answer key

1. lcm(767, 91) = 5369
2. gcd(62, 44) = 2
3. x = 5
4. 33⁻¹ mod 71 = 28
5. 66¹⁷⁴ mod 73 = 46
6. x = 3

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