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

Use Euclid’s algorithm to find gcd(357, 147).

a = 357b = 147

gcd(357, 147) =

2.Easy

Use Euclid’s algorithm to find gcd(900, 165).

a = 900b = 165

gcd(900, 165) =

3.Medium

Solve 29x ≡ 30 (mod 47). Give the solution x with 0 < x < 47.

29x ≡ 30 (mod 47)

x =

4.Medium

Find Euler’s totient φ(21), the number of integers from 1 to 21 that are coprime to 21.

n = 21 = 3·7

φ(21) =

5.Hard

29 is prime. Use Fermat’s little theorem to find 2³⁸³⁶⁴ mod 29.

p = 29k = 38364

2³⁸³⁶⁴ mod 29 =

6.Hard

How many solutions N does 10x ≡ 50 (mod 54) have in {0, 1, …, 53}?

10x ≡ 50 (mod 54)

N =

Answer key

1. gcd(357, 147) = 21
2. gcd(900, 165) = 15
3. x = 14
4. φ(21) = 12
5. 2³⁸³⁶⁴ mod 29 = 16
6. N = 2

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