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(1224, 1176).

a = 1224b = 1176

gcd(1224, 1176) =

2.Easy

Use Euclid’s algorithm to find gcd(636, 432).

a = 636b = 432

gcd(636, 432) =

3.Medium

Solve 42x ≡ 34 (mod 59). Give the solution x with 0 < x < 59.

42x ≡ 34 (mod 59)

x =

4.Medium

Find the inverse of 5 modulo 37: the number x with 0 < x < 37 and 5·x ≡ 1 (mod 37).

a = 5n = 37

5⁻¹ mod 37 =

5.Hard

Use repeated squaring to find 91¹⁸³ mod 97.

a = 91k = 183n = 97

91¹⁸³ mod 97 =

6.Hard

Solve 24x ≡ 14 (mod 34). Give the smallest positive solution x.

24x ≡ 14 (mod 34)

x =

Answer key

1. gcd(1224, 1176) = 24
2. gcd(636, 432) = 12
3. x = 57
4. 5⁻¹ mod 37 = 15
5. 91¹⁸³ mod 97 = 75
6. x = 2

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