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
aqbr104=?·62+??=?·?+??=?·?+??=?·?+?

Find the least common multiple lcm(104, 62). Use Euclid’s algorithm for the gcd first.

a = 104b = 62
lcm(104, 62) =
2.Easy
aqbr328=?·168+??=?·?+??=?·?+?

Use Euclid’s algorithm to find gcd(328, 168).

a = 328b = 168
gcd(328, 168) =
3.Medium

Find the smallest positive integer x with x ≡ 2 (mod 4) and x ≡ 5 (mod 11).

x ≡ 2 (mod 4)x ≡ 5 (mod 11)
x =
4.Medium

Solve 6x ≡ 10 (mod 11). Give the solution x with 0 < x < 11.

6x ≡ 10 (mod 11)
x =
5.Hard

Use repeated squaring to find 723 mod 23.

a = 7k = 23n = 23
723 mod 23 =
6.Hard

How many solutions N does 58x ≡ 48 (mod 90) have in {0, 1, …, 89}?

58x ≡ 48 (mod 90)
N =

Answer key

  1. 1. lcm(104, 62) = 3224
  2. 2. gcd(328, 168) = 8
  3. 3. x = 38
  4. 4. x = 9
  5. 5. 723 mod 23 = 7
  6. 6. N = 2

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