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 (−75) mod 11, the remainder r with 0 ≤ r < 11.

a = −75n = 11

(−75) mod 11 =

2.Easy

Use Euclid’s algorithm to find gcd(441, 297).

a = 441b = 297

gcd(441, 297) =

3.Medium

Find the inverse of 13 modulo 35: the number x with 0 < x < 35 and 13·x ≡ 1 (mod 35).

a = 13n = 35

13⁻¹ mod 35 =

4.Medium

Find the smallest positive integer x with x ≡ 9 (mod 12) and x ≡ 10 (mod 13).

x ≡ 9 (mod 12)x ≡ 10 (mod 13)

x =

5.Hard

61 is prime. Use Fermat’s little theorem to find 17⁹⁴²⁴⁸ mod 61.

p = 61k = 94248

17⁹⁴²⁴⁸ mod 61 =

6.Hard

Solve 40x ≡ 8 (mod 44). Give the smallest positive solution x.

40x ≡ 8 (mod 44)

x =

Answer key

1. (−75) mod 11 = 2
2. gcd(441, 297) = 9
3. 13⁻¹ mod 35 = 27
4. x = 153
5. 17⁹⁴²⁴⁸ mod 61 = 58
6. x = 9

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