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Chapter 4: Problem 7

is there a \(b \in \mathbb{Z}\) such that \(6 b=1 \bmod 81 ?\)

Short Answer

Expert verified
No, there is no integer \(b\) since the GCD of 6 and 81 is not 1.

Step by step solution

01

Understanding the Problem

We need to find if there is an integer value for \(b\) such that \(6b \equiv 1 \pmod{81}\). This means finding a solution to the equation \(6b - 81k = 1\) for some integer \(k\) using the definition of congruence.
02

Finding the GCD Using Euclidean Algorithm

To solve \(6b \equiv 1 \pmod{81}\), first check if the greatest common divisor (GCD) of 6 and 81 is 1. Use the Euclidean algorithm: 1. \(81 = 6 \times 13 + 3\) 2. \(6 = 3 \times 2 + 0\)The GCD of 6 and 81 is 3.
03

Analyzing the GCD

Since the GCD of 6 and 81 is 3, not 1, the equation \(6b \equiv 1 \pmod{81}\) has no solutions, because 6 and 81 are not coprime. A modular inverse exists only if the GCD is 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Euclidean Algorithm
The Euclidean Algorithm is a remarkable tool for finding the greatest common divisor (GCD) of two numbers. It's based on the principle that the GCD of two numbers also divides their difference. Here's how it works:
To find the GCD of two numbers, say 81 and 6, you perform repeated division.
  • Divide 81 by 6, which gives a quotient of 13 and a remainder of 3.
  • Then, replace 81 with 6, and 6 with 3, repeating the process.
  • Now, divide 6 by 3. The remainder is 0, signifying that the last non-zero remainder, which is 3, is the GCD.
The process works efficiently even for large numbers and is key to simplifying many tasks in number theory. Understanding this algorithm is fundamental, especially when dealing with congruences and modular arithmetic in equations.
Congruences
In mathematics, particularly number theory, congruences are a way to express that two numbers are equal modulo some number. For example, when we say \(6b \equiv 1 \pmod{81}\), it means that the difference \(6b - 1\) is divisible by 81. This concept is crucial in solving equations where exact equality isn't required, but rather a relationship modulo a number.
Congruences are often used in coding theory and cryptography. One key aspect is:
  • If two numbers are congruent modulo a number, adding or subtracting the same number to both will not change their congruence.
  • Multiplication works similarly, provided the multiplier and modulus are coprime.
Understanding these properties helps solve many practical problems involving integer equations.
Greatest Common Divisor
The greatest common divisor, or GCD, is the largest integer that divides two numbers without leaving a remainder. It's a fundamental concept in number theory. The Euclidean Algorithm is one of the most efficient ways to find the GCD.
Understanding the GCD is vital since it helps determine if two numbers are coprime鈥攎eaning their GCD is 1. In modular arithmetic, when looking for a solution like \(6b \equiv 1 \pmod{81}\), finding whether a modular inverse exists requires checking if the GCD of the coefficients (6 and 81 in this case) is 1.
  • If the GCD is 1, a solution exists, and a modular inverse can be found.
  • If the GCD is greater than 1, no such inverse exists unless the GCD divides the constant term too.
Hence, determining the GCD isn't just an academic exercise; it assists in practical problem-solving across numerous mathematical fields.

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Most popular questions from this chapter

Compute \(15^{-1}\) mod 19 via Euclid and via Fermat.

Let \(g=x^{5}+x+1 \in \mathrm{F}_{2}[x]\). For each of the two polynomials (i) \(f=x^{3}+x+1 . \quad\) (ii) \(f=x^{3}+1\) in \(F_{2}\left\\{x \mid\right.\), do the following. If \(f \bmod g\) is a unit in \(\left.F_{2} \mid x\right] /\langle g\rangle\), compute its inverse \(h \bmod g\). If \(f\) mod \(g\). is a zero divisor, find a polynomial \(\left.h \in \mathbb{F}_{2} \mid x\right]\) of degree less than 5 such that \(f h \equiv 0\) mod \(g\).

Let \(R\) be a Euclidean domain and \(a, b, c \in R\). (i) Show that the congruence \(a x=b \bmod c\) has a solution \(x \in R\) if and only if \(g=\operatorname{ged}(a, c)\) divides \(b\). Prove that in the latter case, the congruence is equivalent to \((a / g) x=(b / g) x \bmod (c / g)\). (ii) For \(R=Z\) and \(a=5,6,7\), determine whether the congruence \(a x=9\) mod 15 is solvable, and if so, give all solutions \(x \in\\{0, \ldots, 14\\}\).

(i) Let \(a \in \mathbb{N}\) be such that \(0 \leq a<1000\) and the three least significant digits in the decimal representation of \(17 a\) are 001 . What is \(a\) ? (ii) Same question when the least significant digits are 209 .

(i) Expand the rational fractions \(14 / 3\) and \(3 / 14\) into finite continued fractions. (ii) Convert \([2,1,4]\) and \([0,1,1,100]\) into rational numbers.
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