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
aqbr279=?·261+??=?·?+??=?·?+?

Use Euclid’s algorithm to find gcd(279, 261).

a = 279b = 261
gcd(279, 261) =
2.Easy
0123456mod 7

Find (−69) mod 7, the remainder r with 0 ≤ r < 7.

a = −69n = 7
(−69) mod 7 =
3.Medium

Find Euler’s totient φ(85), the number of integers from 1 to 85 that are coprime to 85.

n = 85 = 5·17
φ(85) =
4.Medium

Find the inverse of 61 modulo 79: the number x with 0 < x < 79 and 61·x ≡ 1 (mod 79).

a = 61n = 79
61−1 mod 79 =
5.Hard

Use repeated squaring to find 29199 mod 96.

a = 29k = 199n = 96
29199 mod 96 =
6.Hard
mc = me​ mod ncm = cd​ mod nm(n, e) = (403, 17)d = ?

An RSA key uses the primes p = 13 and q = 31 and the public exponent e = 17. Find the private exponent d (0 < d < φ(n)).

p = 13q = 31e = 17
d =

Answer key

  1. 1. gcd(279, 261) = 9
  2. 2. (−69) mod 7 = 1
  3. 3. φ(85) = 64
  4. 4. 61−1 mod 79 = 57
  5. 5. 29199 mod 96 = 53
  6. 6. d = 233

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