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 152² mod 12 without multiplying out the large numbers: reduce first.

n = 12
152² mod 12 =
2.Easy
aqbr76=?·52+??=?·?+??=?·?+?

Use Euclid’s algorithm to find gcd(76, 52).

a = 76b = 52
gcd(76, 52) =
3.Medium

Solve 6x ≡ 1 (mod 11). Give the solution x with 0 < x < 11.

6x ≡ 1 (mod 11)
x =
4.Medium

Solve 39x ≡ 10 (mod 71). Give the solution x with 0 < x < 71.

39x ≡ 10 (mod 71)
x =
5.Hard

Solve 48x ≡ 42 (mod 78). Give the smallest positive solution x.

48x ≡ 42 (mod 78)
x =
6.Hard

Use repeated squaring to find 2925 mod 40.

a = 29k = 25n = 40
2925 mod 40 =

Answer key

  1. 1. 152² mod 12 = 4
  2. 2. gcd(76, 52) = 4
  3. 3. x = 2
  4. 4. x = 13
  5. 5. x = 9
  6. 6. 2925 mod 40 = 29

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