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 130 mod 9, the remainder r with 0 ≤ r < 9.

a = 130n = 9

130 mod 9 =

2.Easy

Find 330² mod 13 without multiplying out the large numbers: reduce first.

n = 13

330² mod 13 =

3.Medium

Find Euler’s totient φ(19), the number of integers from 1 to 19 that are coprime to 19.

n = 19 = 19

φ(19) =

4.Medium

Solve 25x ≡ 37 (mod 67). Give the solution x with 0 < x < 67.

25x ≡ 37 (mod 67)

x =

5.Hard

An RSA key uses the primes p = 19 and q = 37 and the public exponent e = 5. Find the private exponent d (0 < d < φ(n)).

p = 19q = 37e = 5

d =

6.Hard

53 is prime. Use Fermat’s little theorem to find 22⁸⁵⁵⁵⁸ mod 53.

p = 53k = 85558

22⁸⁵⁵⁵⁸ mod 53 =

Answer key

1. 130 mod 9 = 4
2. 330² mod 13 = 12
3. φ(19) = 18
4. x = 39
5. d = 389
6. 22⁸⁵⁵⁵⁸ mod 53 = 43

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