10.9. Finding Solutions in a Specific Range

Try these first

1. 1.

   What is the solution to 2x ≡ 4 (mod 3)?

   Hint

   Find the integer value of `x` satisfying the equation.

2. 2.

   Determine if 1x ≡ 5 (mod 10) has solutions, and if so, provide one.

   Hint

   Solve the equation normally then check for modulo.

3. 3.

   What kind of equations are expressed as ax ≡ b (mod N)?

   • Linear equations
   • Linear Congruences
   • Quadratic equations

   Hint

   Look for terms that involve modulus.

4. 4.

   True or False: Every linear congruence has at least one solution.

   • True
   • False

   Hint

   Consider the structure of congruences.

5. 5.

   Prove that if ax ≡ b (mod N) has a solution for GCD(a, N) = d, then it implies a system of equations.

   Hint

   Use properties of divisors and modular arithmetic.

6. 6.

   For three congruences x ≡ 1 (mod 6), x ≡ 2 (mod 15), and x ≡ 3 (mod 10), find all solutions up to M.

   Hint

   Work through each congruence step to consolidate solutions.