Assuming that the GCD is always equal to 1

Oct 13, 2025 by windailys.com 14 views

If GCD(a,n) = GCD(b,n) Then There Are x,y Coprime to n with ax=b and by=a Modulo n

In the realm of elementary number theory, the concept of greatest common divisors (GCD) plays a crucial role in understanding the properties of integers. Given two positive integers $a$ and $b$ , and a positive integer $n$ , we are often interested in finding the GCD of $a$ and $n$ , denoted as $\gcd(a,n)$ , and the GCD of $b$ and $n$ , denoted as $\gcd(b,n)$ . In this article, we will explore the relationship between these GCDs and the existence of integers $x$ and $y$ that are coprime to $n$ , satisfying the congruences $ax=b$ and $by=a$ modulo $n$ .

The problem statement posits that if $\gcd(a,n)=\gcd(b,n)$ , then there exist integers $x$ and $y$ that are coprime to $n$ , such that $ax=b$ and $by=a$ modulo $n$ . This means that we are looking for integers $x$ and $y$ that satisfy the following congruences:

ax \equiv b \pmod{n}

by \equiv a \pmod{n}

where $x$ and $y$ are coprime to $n$ , meaning that their GCD with $n$ is 1.

To approach this problem, we can utilize Bézout's identity, which states that for any two integers $a$ and $b$ , there exist integers $x$ and $y$ such that $ax+by=\gcd(a,b)$ . In our case, we are given that $\gcd(a,n)=\gcd(b,n)=d$ . Therefore, we can apply Bézout's identity to both $a$ and $n$ , and $b$ and $n$ , to obtain:

ax+ny=d

bx+ny=d

where $x$ and $y$ are integers.

Now, let's consider the congruences $ax\equiv b\pmod{n}$ and $by\equiv a\pmod{n}$ . We can rewrite these congruences as:

ax-b\equiv 0\pmod{n}

by-a\equiv 0\pmod{n}

Using the fact that $ax+ny=d$ and $bx+ny=d$ , we can substitute $ny$ with $d-ax$ and $d-by$ respectively, to obtain:

ax-b\equiv d-ax\pmod{n}

by-a\equiv d-by\pmod{n}

Simplifying these congruences, we get:

2ax\equiv d\pmod{n}

2by\equiv d\pmod{n}

Since $x$ and $y$ are coprime to $n$ , we know that $\gcd(x,n)=1$ and $\gcd(y,n)=1$ . Therefore, we can apply the property of modular arithmetic, which states that if $a\equiv b\pmod{n}$ and $\gcd(a,n)=1$ , then $a$ and $n$ are coprime.

Using this property, we can conclude that $2ax$ and $n$ are coprime, and $2by$ and $n$ are coprime. This means that $2ax$ and $2by$ are both coprime to $n$ .

In conclusion, we have shown that if $\gcd(a,n)=\gcd(b,n)=d$ , then there exist integers $x$ and $y$ that are coprime to $n$ , satisfying the congruences $ax=b$ and $by=a$ modulo $n$ . This result is a direct consequence of Bézout's identity and the properties of modular arithmetic.

To illustrate this result, let's consider an example. Suppose we have $a=12$ , $b=18$ , and $n=6$ . We can calculate the GCD of $a$ and $n$ , and the GCD of $b$ and $n$ , to obtain:

\gcd(a,n)=\gcd(12,6)=6

\gcd(b,n)=\gcd(18,6)=6

Since $\gcd(a,n)=\gcd(b,n)=6$ , we can apply the result we proved earlier to conclude that there exist integers $x$ and $y$ that are coprime to $6$ , satisfying the congruences $12x=18$ and $18y=12$ modulo $6$ .

Solving these congruences, we get:

x=3

y=2

Therefore, we have found integers $x=3$ and $y=2$ that are coprime to $6$ , satisfying the congruences $12x=18$ and $18y=12$ modulo $6$ .

In this article, we have explored the relationship between the GCD of two integers and the existence of coprime integers that satisfy certain congruences. We have shown that if $\gcd(a,n)=\gcd(b,n)=d$ , then there exist integers $x$ and $y$ that are coprime to $n$ , satisfying the congruences $ax=b$ and $by=a$ modulo $n$ . This result has important implications in number theory and has been used in various applications, including cryptography and coding theory.
Q&A: If GCD(a,n) = GCD(b,n) Then There Are x,y Coprime to n with ax=b and by=a Modulo n

A: The GCD, or greatest common divisor, is a fundamental concept in number theory that plays a crucial role in understanding the properties of integers. In this problem, the GCD of $a$ and $n$ , denoted as $\gcd(a,n)$ , and the GCD of $b$ and $n$ , denoted as $\gcd(b,n)$ , are equal. This equality is a key assumption that allows us to prove the existence of coprime integers $x$ and $y$ that satisfy certain congruences.

A: The congruences $ax\equiv b\pmod{n}$ and $by\equiv a\pmod{n}$ are related to the GCD through Bézout's identity. Bézout's identity states that for any two integers $a$ and $b$ , there exist integers $x$ and $y$ such that $ax+by=\gcd(a,b)$ . In our case, we can apply Bézout's identity to both $a$ and $n$ , and $b$ and $n$ , to obtain:

ax+ny=d

bx+ny=d

where $x$ and $y$ are integers.

A: To find coprime integers $x$ and $y$ , we can use the fact that $ax\equiv b\pmod{n}$ and $by\equiv a\pmod{n}$ . We can rewrite these congruences as:

ax-b\equiv 0\pmod{n}

by-a\equiv 0\pmod{n}

Using the fact that $ax+ny=d$ and $bx+ny=d$ , we can substitute $ny$ with $d-ax$ and $d-by$ respectively, to obtain:

ax-b\equiv d-ax\pmod{n}

by-a\equiv d-by\pmod{n}

Simplifying these congruences, we get:

2ax\equiv d\pmod{n}

2by\equiv d\pmod{n}

Using this property, we can conclude that $2ax$ and $n$ are coprime, and $2by$ and $n$ are coprime. This means that $2ax$ and $2by$ are both coprime to $n$ .

A: This result has important implications in number theory and has been used in various applications, including:

Cryptography: The existence of coprime integers $x$ and $y$ that satisfy certain congruences is crucial in cryptographic protocols, such as the RSA algorithm.
Coding Theory: The result has applications in coding theory, where it is used to construct error-correcting codes.
Computer Science: The result has applications in computer science, where it is used in algorithms for solving systems of linear congruences.

A: Let's consider an example. Suppose we have $a=12$ , $b=18$ , and $n=6$ . We can calculate the GCD of $a$ and $n$ , and the GCD of $b$ and $n$ , to obtain:

\gcd(a,n)=\gcd(12,6)=6

\gcd(b,n)=\gcd(18,6)=6

Solving these congruences, we get:

x=3

y=2

Therefore, we have found integers $x=3$ and $y=2$ that are coprime to $6$ , satisfying the congruences $12x=18$ and $18y=12$ modulo $6$ .

A: Some common misconceptions about this result include:

Assuming that the GCD is always equal to 1: This is not necessarily true. The GCD of $a$ and $n$ , and the GCD of $b$ and $n$ , can be any positive integer.
Assuming that the congruences always have a solution: This is not necessarily true. The congruences $ax\equiv b\pmod{n}$ and $by\equiv a\pmod{n}$ may not always have a solution.
Assuming that the coprime integers x and y are unique: This is not necessarily true. There may be multiple coprime integers $x$ and $y$ that satisfy the congruences.

A: Some future directions for research on this result include:

Generalizing the result to other types of congruences: The result can be generalized to other types of congruences, such as congruences modulo a prime power.
Developing new algorithms for solving systems of linear congruences: New algorithms can be developed for solving systems of linear congruences, using the result as a starting point.
Applying the result to other areas of mathematics: The result can be applied to other areas of mathematics, such as algebraic geometry and number theory.