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 (−138) mod 11, the remainder r with 0 ≤ r < 11.

a = −138n = 11

(−138) mod 11 =

2.Easy

Find 473 mod 6, the remainder r with 0 ≤ r < 6.

a = 473n = 6

473 mod 6 =

3.Medium

Find Euler’s totient φ(26), the number of integers from 1 to 26 that are coprime to 26.

n = 26 = 2·13

φ(26) =

4.Medium

Find the inverse of 5 modulo 29: the number x with 0 < x < 29 and 5·x ≡ 1 (mod 29).

a = 5n = 29

5⁻¹ mod 29 =

5.Hard

43 is prime. Use Fermat’s little theorem to find 11¹³²⁵⁰ mod 43.

p = 43k = 13250

11¹³²⁵⁰ mod 43 =

6.Hard

41 is prime. Use Fermat’s little theorem to find 7³⁹⁷²⁷ mod 41.

p = 41k = 39727

7³⁹⁷²⁷ mod 41 =

Answer key

1. (−138) mod 11 = 5
2. 473 mod 6 = 5
3. φ(26) = 12
4. 5⁻¹ mod 29 = 6
5. 11¹³²⁵⁰ mod 43 = 4
6. 7³⁹⁷²⁷ mod 41 = 17

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