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 (295·184 + 226) mod 10 without multiplying out the large numbers: reduce first.

n = 10
(295·184 + 226) mod 10 =
2.Easy

Find 845² mod 6 without multiplying out the large numbers: reduce first.

n = 6
845² mod 6 =
3.Medium

Find the smallest positive integer x with x ≡ 8 (mod 9) and x ≡ 10 (mod 11).

x ≡ 8 (mod 9)x ≡ 10 (mod 11)
x =
4.Medium

Find the smallest positive integer x with x ≡ 3 (mod 5) and x ≡ 5 (mod 13).

x ≡ 3 (mod 5)x ≡ 5 (mod 13)
x =
5.Hard

Use repeated squaring to find 1740 mod 67.

a = 17k = 40n = 67
1740 mod 67 =
6.Hard

Use repeated squaring to find 22128 mod 37.

a = 22k = 128n = 37
22128 mod 37 =

Answer key

  1. 1. (295·184 + 226) mod 10 = 6
  2. 2. 845² mod 6 = 1
  3. 3. x = 98
  4. 4. x = 18
  5. 5. 1740 mod 67 = 54
  6. 6. 22128 mod 37 = 34

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