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
aqbr216=?·188+??=?·?+??=?·?+??=?·?+??=?·?+?

Find the least common multiple lcm(216, 188). Use Euclid’s algorithm for the gcd first.

a = 216b = 188
lcm(216, 188) =
2.Easy

Find (698·440 + 269) mod 13 without multiplying out the large numbers: reduce first.

n = 13
(698·440 + 269) mod 13 =
3.Medium

Find the inverse of 59 modulo 90: the number x with 0 < x < 90 and 59·x ≡ 1 (mod 90).

a = 59n = 90
59−1 mod 90 =
4.Medium

Find the inverse of 62 modulo 85: the number x with 0 < x < 85 and 62·x ≡ 1 (mod 85).

a = 62n = 85
62−1 mod 85 =
5.Hard

Use repeated squaring to find 67191 mod 80.

a = 67k = 191n = 80
67191 mod 80 =
6.Hard

Solve 46x ≡ 26 (mod 52). Give the smallest positive solution x.

46x ≡ 26 (mod 52)
x =

Answer key

  1. 1. lcm(216, 188) = 10152
  2. 2. (698·440 + 269) mod 13 = 4
  3. 3. 59−1 mod 90 = 29
  4. 4. 62−1 mod 85 = 48
  5. 5. 67191 mod 80 = 43
  6. 6. x = 13

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