We are tasked with solving the congruence:

\[
126x \equiv 46 \ (\text{mod} \ 99)
\]

This is a linear congruence of the form \( ax \equiv b \ (\text{mod} \ m) \), where \( a = 126 \), \( b = 46 \), and \( m = 99 \). To solve for \( x \), follow these steps:

### Step 1: Simplify the coefficients

The greatest common divisor (gcd) of 126 and 99 is:

\[
\text{gcd}(126, 99) = 9
\]

Since the gcd divides both 126 and 99, we can simplify the equation by dividing through by 9. This leads to the new congruence:

\[
\frac{126}{9}x \equiv \frac{46}{9} \ (\text{mod} \ \frac{99}{9})
\]

Simplifying this, we get:

\[
14x \equiv 5 \ (\text{mod} \ 11)
\]

### Step 2: Solve the new congruence

Now, we need to solve \( 14x \equiv 5 \ (\text{mod} \ 11) \). First, reduce \( 14 \ (\text{mod} \ 11) \) to get:

\[
14 \equiv 3 \ (\text{mod} \ 11)
\]

So, the congruence becomes:

\[
3x \equiv 5 \ (\text{mod} \ 11)
\]

### Step 3: Find the multiplicative inverse of 3 mod 11

To solve \( 3x \equiv 5 \ (\text{mod} \ 11) \), we need the inverse of 3 modulo 11. We can find it using the Extended Euclidean Algorithm. Applying the algorithm:

\[
11 = 3(3) + 2
\]
\[
3 = 2(1) + 1
\]
\[
2 = 1(2) + 0
\]

Back-substitute to express 1 as a combination of 3 and 11:

\[
1 = 3 - 1(2)
\]
\[
1 = 3 - 1(11 - 3(3)) = 4(3) - 1(11)
\]

Thus, \( 3^{-1} \equiv 4 \ (\text{mod} \ 11) \).

### Step 4: Solve for \( x \)

Now, multiply both sides of \( 3x \equiv 5 \ (\text{mod} \ 11) \) by 4, the inverse of 3:

\[
x \equiv 4(5) \ (\text{mod} \ 11)
\]
\[
x \equiv 20 \ (\text{mod} \ 11)
\]
\[
x \equiv 9 \ (\text{mod} \ 11)
\]

### Step 5: Back-substitute to the original modulus

Since \( x \equiv 9 \ (\text{mod} \ 11) \), the general solution for \( x \) in the original modulus 99 is given by:

\[
x = 9 + 11k
\]

Thus, \( x = 9, 20, 31, \dots \).

### Final Answer

The solution to the original congruence \( 126x \equiv 46 \ (\text{mod} \ 99) \) is:

\[
x \equiv 9 \ (\text{mod} \ 11)
\]

Let me know if you'd like further explanation or a detailed step breakdown.

---

Relative questions:
1. How is the Extended Euclidean Algorithm used to find inverses in modular arithmetic?
2. What are linear congruences and how do they differ from other types of congruences?
3. What is the significance of the greatest common divisor in solving modular equations?
4. How do modular inverses work and why are they important in solving congruences?
5. Can you explain why we can divide both sides of a congruence by the gcd?

Tip: Always simplify the coefficients of a congruence when the gcd of the modulus and coefficient isn't 1.

Learn how to solve the linear congruence 126x ≡ 46 (mod 99) using modular arithmetic, GCD, and the Extended Euclidean Algorithm. Follow the step-by-step solution to understand how to find the multiplicative inverse and simplify the problem for advanced high school students.

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