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
012345mod 6

Find (−141) mod 6, the remainder r with 0 ≤ r < 6.

a = −141n = 6
(−141) mod 6 =
2.Easy

Find (956·651 + 47) mod 10 without multiplying out the large numbers: reduce first.

n = 10
(956·651 + 47) mod 10 =
3.Medium

Find Euler’s totient φ(24), the number of integers from 1 to 24 that are coprime to 24.

n = 24 = 23·3
φ(24) =
4.Medium

Find the smallest positive integer x with x ≡ 1 (mod 9) and x ≡ 0 (mod 11).

x ≡ 1 (mod 9)x ≡ 0 (mod 11)
x =
5.Hard

97 is prime. Use Fermat’s little theorem to find 594588 mod 97.

p = 97k = 94588
594588 mod 97 =
6.Hard

83 is prime. Use Fermat’s little theorem to find 1737711 mod 83.

p = 83k = 37711
1737711 mod 83 =

Answer key

  1. 1. (−141) mod 6 = 3
  2. 2. (956·651 + 47) mod 10 = 3
  3. 3. φ(24) = 8
  4. 4. x = 55
  5. 5. 594588 mod 97 = 73
  6. 6. 1737711 mod 83 = 63

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