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(240, 196).

a = 240b = 196

gcd(240, 196) =

2.Easy

Find (570·711 + 88) mod 5 without multiplying out the large numbers: reduce first.

n = 5

(570·711 + 88) mod 5 =

3.Medium

Find the smallest positive integer x with x ≡ 4 (mod 8) and x ≡ 8 (mod 13).

x ≡ 4 (mod 8)x ≡ 8 (mod 13)

x =

4.Medium

Find the inverse of 20 modulo 67: the number x with 0 < x < 67 and 20·x ≡ 1 (mod 67).

a = 20n = 67

20⁻¹ mod 67 =

5.Hard

53 is prime. Use Fermat’s little theorem to find 3⁶³⁰⁹⁸ mod 53.

p = 53k = 63098

3⁶³⁰⁹⁸ mod 53 =

6.Hard

73 is prime. Use Fermat’s little theorem to find 19⁵⁵²⁹³ mod 73.

p = 73k = 55293

19⁵⁵²⁹³ mod 73 =

Answer key

1. gcd(240, 196) = 4
2. (570·711 + 88) mod 5 = 3
3. x = 60
4. 20⁻¹ mod 67 = 57
5. 3⁶³⁰⁹⁸ mod 53 = 17
6. 19⁵⁵²⁹³ mod 73 = 24

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