Math Problem Statement

We are now looking at congruences rather than equations. So we want the solution to be in the form of a congruence as well. For example, for 8(b), your solution for x should be of the form x ≡ a(mod 50). You can double check that your solution is correct by calculating 13(a) mod 50 and it should be 42.

For each of the following linear congruences, find all of its solutions. (a) 15x ≡ 9 (mod 25) (b) 13x ≡ 42 (mod 50)

Solution

Let's solve each linear congruence in the required form.

Part (a): 15x≡9(mod25)15x \equiv 9 \pmod{25}

To solve this, we need to find all values of xx that satisfy: 15x≡9(mod25).15x \equiv 9 \pmod{25}.

  1. First, let's check if there is a solution by examining the gcd of 15 and 25.

    • gcd⁡(15,25)=5\gcd(15, 25) = 5.
    • Since 5 divides the right-hand side (9), a solution exists.

  2. Simplify the congruence by dividing everything by the gcd (5): 3x≡95(mod255).3x \equiv \frac{9}{5} \pmod{\frac{25}{5}}.

    • Simplified equation: [ 3x \equiv 9 \pmod{5}.

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Math Problem Analysis

Mathematical Concepts

Linear Congruences
Number Theory
Modular Arithmetic

Formulas

ax ≡ b (mod m)

Theorems

Existence of solutions in linear congruences
Greatest Common Divisor (gcd) and divisibility

Suitable Grade Level

Grades 10-12

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