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 132² mod 7 without multiplying out the large numbers: reduce first.

n = 7
132² mod 7 =
2.Easy

Find (642·511) mod 11 without multiplying out the large numbers: reduce first.

n = 11
(642·511) mod 11 =
3.Medium

Find the smallest positive integer x with x ≡ 1 (mod 5) and x ≡ 4 (mod 7).

x ≡ 1 (mod 5)x ≡ 4 (mod 7)
x =
4.Medium

Find the inverse of 67 modulo 73: the number x with 0 < x < 73 and 67·x ≡ 1 (mod 73).

a = 67n = 73
67−1 mod 73 =
5.Hard

37 is prime. Use Fermat’s little theorem to find 1750362 mod 37.

p = 37k = 50362
1750362 mod 37 =
6.Hard

41 is prime. Use Fermat’s little theorem to find 2818533 mod 41.

p = 41k = 18533
2818533 mod 41 =

Answer key

  1. 1. 132² mod 7 = 1
  2. 2. (642·511) mod 11 = 9
  3. 3. x = 11
  4. 4. 67−1 mod 73 = 12
  5. 5. 1750362 mod 37 = 21
  6. 6. 2818533 mod 41 = 30

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