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Chapter 4: Problem 18

If \(a \equiv b\left(\bmod n_{1}\right)\) and \(a \equiv c\left(\bmod n_{2}\right)\), prove that \(b \equiv c(\bmod n)\), where the integer \(n=\operatorname{gcd}\left(n_{1}, n_{2}\right)\)

Short Answer

Expert verified
\(b - c\) is a multiple of \(n\), so \(b \equiv c(\bmod n)\).

Step by step solution

01

Understanding Given Congruences

We need to prove that if \(a \equiv b(\bmod n_1)\) and \(a \equiv c(\bmod n_2)\), then \(b \equiv c(\bmod n)\), where \(n = \gcd(n_1, n_2)\). This means \(a\) differs from \(b\) by a multiple of \(n_1\), and from \(c\) by a multiple of \(n_2\).
02

Expressing Congruences as Equations

From \(a \equiv b(\bmod n_1)\), we have \(a = b + k_1 n_1\) for some integer \(k_1\). Similarly, from \(a \equiv c(\bmod n_2)\), we have \(a = c + k_2 n_2\) for some integer \(k_2\).
03

Equating the Expressions for \(a\)

Since both expressions equal \(a\), we can set them equal to each other: \[b + k_1 n_1 = c + k_2 n_2.\] Rearrange this to get \[b - c = k_2 n_2 - k_1 n_1.\]
04

Introducing the GCD Concept

The equation \(b - c = k_2 n_2 - k_1 n_1\) implies that \(b - c\) is a linear combination of \(n_1\) and \(n_2\). By the definition of the greatest common divisor, \(n = \gcd(n_1, n_2)\) divides any such linear combination. Thus, \(b - c\) is divisible by \(n\).
05

Conclusion: Proving the Result

Since \(n\) divides \(b - c\), we establish that \(b \equiv c(\bmod n)\). Therefore, the congruence \(b \equiv c(\bmod n)\) holds true as required.

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Key Concepts

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

Congruence Relations
In the realm of number theory, congruence relations create a bridge between numbers through a certain modulus. They can be thought of as a kind of arithmetic "equivalence." When we write \( a \equiv b \pmod{n} \), we are expressing that \( a - b \) is a multiple of \( n \). What does this mean? It means when you subtract \( b \) from \( a \), the result is divisible by \( n \).

In the context of the given exercise, the expressions \( a \equiv b(\bmod n_1) \) and \( a \equiv c(\bmod n_2) \) tell us that both distances between \( a \) and \( b \), and \( a \) and \( c \), are tied to different multiples of numbers \( n_1 \) and \( n_2 \).
  • This forms the basis for proving relationships between numbers under different moduli.
  • Understanding this basic relationship helps in proving deeper properties, like connecting different congruences to find a common ground.
Greatest Common Divisor
The greatest common divisor, or GCD, of two integers \( n_1 \) and \( n_2 \), is the largest number that divides both \( n_1 \) and \( n_2 \) without leaving any remainder. This pivotal concept serves as a cornerstone for many proofs and computations in number theory.

In the exercise, when we declare \( n = \gcd(n_1, n_2) \), we're identifying a number that captures the shared divisibility of both \( n_1 \) and \( n_2 \). This relationship is essential when we explore the potential equivalence between other numbers by showing they can be expressed as a linear combination involving the GCD.
  • Why is this important here? Because it ensures that any common factor of \( n_1 \) and \( n_2 \) will also divide any linear combination of these numbers.
  • This tells us that the difference \( b-c \) fits neatly into a framework divided evenly by the GCD, establishing \( b \equiv c \pmod{n} \).
Linear Combinations
Linear combinations form a foundational concept in algebra and number theory. Simply put, to form a linear combination of two numbers \( n_1 \) and \( n_2 \), you take integer multiples of each number and sum them up.

Mathematically, this looks like \( k_1 n_1 + k_2 n_2 \), where \( k_1 \) and \( k_2 \) are integers. In the context of the exercise, the equation \( b - c = k_2 n_2 - k_1 n_1 \) exemplifies this idea. Here, \( b - c \) is indeed a linear combination of \( n_1 \) and \( n_2 \).
  • The equation helps translate the abstract rule of congruences into an understandable number form.
  • When the GCD divides this combination, it confirms our congruence relationship because any linear combination's divisor relationship is grounded in the GCD.
  • Linear combinations assure that relationships between numbers hold consistently under different operations like addition and subtraction, deepening the proofs of congruence.

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Most popular questions from this chapter

If \(p\) is a prime satisfying \(n

Solve the following linear congruences: (a) \(25 x=15(\bmod 29)\). (b) \(5 x \equiv 2(\bmod 26)\) (c) \(6 x=15(\bmod 21)\). (d) \(36 x=8(\) mod 102\()\) (c) \(34 x \equiv 60(\bmod 98)\). (f) \(140 x \equiv 133(\bmod 301)\). [Hint: \(\operatorname{gcd}(140,301)=7 .]\)

Prove the following statements: (a) For any integer \(a\), the units digit of \(a^{2}\) is \(0,1,4,5,6\), or \(9 .\) (b) Any one of the integers \(0,1,2,3,4,5,6,7,8,9\) can occur as the units digit of \(a^{3}\). (c) For any integer \(a\), the units digit of \(a^{4}\) is \(0,1,5\), or 6 . (d) The units digit of a triangular number is \(0,1,3,5,6\), or 8 .

(a) Given an integer \(N\), let \(M\) be the integer formed by reversing the order of the digits of \(N\) (for example, if \(N=6923\), then \(M=3296\) ). Verify that \(N-M\) is divisible by \(9 .\) (b) A palindrome is a number that reads the same backwards as forwards (for instance, 373 and 521125 are palindromes). Prove that any palindrome with an even number of digits is divisible by 11 .

Verify that \(0,1,2,2^{2}, 2^{3}, \ldots .2^{9}\) form a complete set of residues modulo 11 , but that \(0,1^{2}, 2^{2}, 3^{2}, \ldots, 10^{2}\) do not.
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