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
aqbr344=?·152+??=?·?+??=?·?+??=?·?+?

Use Euclid’s algorithm to find gcd(344, 152).

a = 344b = 152
gcd(344, 152) =
2.Easy

Find (694·657) mod 8 without multiplying out the large numbers: reduce first.

n = 8
(694·657) mod 8 =
3.Medium

Find the inverse of 12 modulo 19: the number x with 0 < x < 19 and 12·x ≡ 1 (mod 19).

a = 12n = 19
12−1 mod 19 =
4.Medium

Find the smallest positive integer x with x ≡ 5 (mod 7) and x ≡ 3 (mod 12).

x ≡ 5 (mod 7)x ≡ 3 (mod 12)
x =
5.Hard

How many solutions N does 10x ≡ 2 (mod 12) have in {0, 1, …, 11}?

10x ≡ 2 (mod 12)
N =
6.Hard

Solve 44x ≡ 38 (mod 70). Give the smallest positive solution x.

44x ≡ 38 (mod 70)
x =

Answer key

  1. 1. gcd(344, 152) = 8
  2. 2. (694·657) mod 8 = 6
  3. 3. 12−1 mod 19 = 8
  4. 4. x = 75
  5. 5. N = 2
  6. 6. x = 12

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