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 (896·34) mod 12 without multiplying out the large numbers: reduce first.

n = 12
(896·34) mod 12 =
2.Easy
012345678910mod 11

Find (−58) mod 11, the remainder r with 0 ≤ r < 11.

a = −58n = 11
(−58) mod 11 =
3.Medium

Find Euler’s totient φ(25), the number of integers from 1 to 25 that are coprime to 25.

n = 25 = 52
φ(25) =
4.Medium

Find Euler’s totient φ(98), the number of integers from 1 to 98 that are coprime to 98.

n = 98 = 2·72
φ(98) =
5.Hard

How many solutions N does 4x ≡ 8 (mod 30) have in {0, 1, …, 29}?

4x ≡ 8 (mod 30)
N =
6.Hard

How many solutions N does 70x ≡ 34 (mod 74) have in {0, 1, …, 73}?

70x ≡ 34 (mod 74)
N =

Answer key

  1. 1. (896·34) mod 12 = 8
  2. 2. (−58) mod 11 = 8
  3. 3. φ(25) = 20
  4. 4. φ(98) = 42
  5. 5. N = 2
  6. 6. N = 2

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