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(748, 357).

a = 748b = 357

gcd(748, 357) =

2.Easy

Find 938 mod 9, the remainder r with 0 ≤ r < 9.

a = 938n = 9

938 mod 9 =

3.Medium

Find the smallest positive integer x with x ≡ 5 (mod 7) and x ≡ 10 (mod 12).

x ≡ 5 (mod 7)x ≡ 10 (mod 12)

x =

4.Medium

Solve 17x ≡ 16 (mod 37). Give the solution x with 0 < x < 37.

17x ≡ 16 (mod 37)

x =

5.Hard

Use repeated squaring to find 40⁵¹ mod 93.

a = 40k = 51n = 93

40⁵¹ mod 93 =

6.Hard

An RSA key uses the primes p = 19 and q = 23 and the public exponent e = 5. Find the private exponent d (0 < d < φ(n)).

p = 19q = 23e = 5

d =

Answer key

1. gcd(748, 357) = 17
2. 938 mod 9 = 2
3. x = 82
4. x = 14
5. 40⁵¹ mod 93 = 70
6. d = 317

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