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 328² mod 13 without multiplying out the large numbers: reduce first.

n = 13

328² mod 13 =

2.Easy

Use Euclid’s algorithm to find gcd(720, 264).

a = 720b = 264

gcd(720, 264) =

3.Medium

Find the smallest positive integer x with x ≡ 7 (mod 9) and x ≡ 6 (mod 10).

x ≡ 7 (mod 9)x ≡ 6 (mod 10)

x =

4.Medium

Solve 13x ≡ 17 (mod 21). Give the solution x with 0 < x < 21.

13x ≡ 17 (mod 21)

x =

5.Hard

23 is prime. Use Fermat’s little theorem to find 21²²⁹⁵⁷ mod 23.

p = 23k = 22957

21²²⁹⁵⁷ mod 23 =

6.Hard

How many solutions N does 12x ≡ 28 (mod 46) have in {0, 1, …, 45}?

12x ≡ 28 (mod 46)

N =

Answer key

1. 328² mod 13 = 9
2. gcd(720, 264) = 24
3. x = 16
4. x = 11
5. 21²²⁹⁵⁷ mod 23 = 22
6. N = 2

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