- Identify the modular inverse of 3 modulo 7 by using the given identity \(3(-2) \equiv 1 \pmod{7}\), which simplifies to the inverse being \(5\).
- Multiply both sides of the congruence \(3x \equiv 5 \pmod{7}\) by \(5\) to isolate \(x\), resulting in \(x \equiv 25 \equiv 4 \pmod{7}\).
- Formulate the general solution using the particular solution \(x=4\) and the modulus \(7\), expressed as \(x = 4 + 7t\) for any integer \(t\).
- The final solution is \(\boxed{x \equiv 4 \pmod{7}}\).
Explanation
- Identify the Modular Inverse
We are given the linear congruence \(3x \equiv 5 \pmod{7}\) and the identity \(3(-2) + 7(1) = 1\). This identity shows that \(3 \times (-2) = -6\), and since \(-6 \equiv 1 \pmod{7}\), the value \(-2\) acts as the modular multiplicative inverse of \(3\) modulo \(7\). We can rewrite \(-2 \pmod{7}\) as \(5\) because \(-2 + 7 = 5\). Thus, \(3 \times 5 = 15 = 2 \times 7 + 1 \equiv 1 \pmod{7}\).
- Solve for x
To solve \(3x \equiv 5 \pmod{7}\), we multiply both sides of the congruence by the modular inverse of \(3\), which we found to be \(5\).
Multiplying by \(5\):
\(5 \cdot (3x) \equiv 5 \cdot 5 \pmod{7}\)
\((5 \cdot 3)x \equiv 25 \pmod{7}\)
Since \(15 \equiv 1 \pmod{7}\) and \(25 \equiv 4 \pmod{7}\) (because \(25 = 3 \times 7 + 4\)):
\(x \equiv 4 \pmod{7}\)
This gives us the smallest non-negative particular solution \(x_0 = 4\).
- Determine the General Solution
The general solution to a linear congruence \(ax \equiv b \pmod{n}\) is given by \(x \equiv x_0 + n \cdot t \pmod{n}\), where \(x_0\) is a particular solution and \(t\) is any integer. Substituting our particular solution \(x_0 = 4\) and the modulus \(n = 7\), we get the general solution:
\(x = 4 + 7t\) for any integer \(t\).
- Final Answer
We have determined that the particular solution is \(4\) and the general solution is \(x = 4 + 7t\).
Final answer: \(x \equiv 4 \pmod{7}\) or \(x = 4 + 7t\). \(\boxed{x \equiv 4 \pmod{7}}\)
Examples
Modular arithmetic is frequently used in computer science and cryptography for generating pseudo-random numbers and securing digital communications. For example, if you are designing a system that assigns unique IDs in a rotating cycle, solving a congruence like \(3x \equiv 5 \pmod{7}\) helps determine exactly which ID number corresponds to a specific step in the sequence. By understanding how to find these solutions, you ensure that your system remains predictable and correctly aligned across all operational cycles.