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 (−148) mod 5, the remainder r with 0 ≤ r < 5.

a = −148n = 5

(−148) mod 5 =

2.Easy

Find 359 mod 3, the remainder r with 0 ≤ r < 3.

a = 359n = 3

359 mod 3 =

3.Medium

Find the inverse of 37 modulo 59: the number x with 0 < x < 59 and 37·x ≡ 1 (mod 59).

a = 37n = 59

37⁻¹ mod 59 =

4.Medium

Find the inverse of 31 modulo 43: the number x with 0 < x < 43 and 31·x ≡ 1 (mod 43).

a = 31n = 43

31⁻¹ mod 43 =

5.Hard

Use repeated squaring to find 65¹⁴⁴ mod 92.

a = 65k = 144n = 92

65¹⁴⁴ mod 92 =

6.Hard

Solve 30x ≡ 36 (mod 63). Give the smallest positive solution x.

30x ≡ 36 (mod 63)

x =

Answer key

1. (−148) mod 5 = 2
2. 359 mod 3 = 2
3. 37⁻¹ mod 59 = 8
4. 31⁻¹ mod 43 = 25
5. 65¹⁴⁴ mod 92 = 73
6. x = 18

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