Question

Show that if \(a, b, m\) and \(n\) are integers such that \(m\) and \(n\) are positive, \(n \mid m\) and \(a \equiv b(\bmod m)\), then \(a \equiv b(\bmod n)\).

Short answer

Question: If \(a, b, m\) and \(n\) are integers such that \(m\) and \(n\) are positive, \(n \mid m\) and \(a \equiv b(\bmod m)\), prove that \(a \equiv b(\bmod n)\).

Step-by-step solution

Recalling the definition of modular congruence

The definition of modular congruence states that the following condition holds: "\(a\) is congruent to \(b\) modulo \(m\) if and only if their difference \(a-b\) is divisible by m." Mathematically speaking, \(a \equiv b(\bmod m)\) if and only if \(m\mid (a-b)\). Similarly, we need to show that \(n\mid (a-b)\) to prove that \(a \equiv b(\bmod n)\).

Use the given conditions

We are given that \(n \mid m\) which means that there exists an integer \(k\) such that \(m = n\cdot k\). Now, let's use the condition of \(a \equiv b(\bmod m)\). According to the definition of modular congruence, this means that \(m\mid (a-b)\). Therefore, there exists an integer \(q\) such that \((a-b) = m\cdot q\).

Express \(a-b\) in terms of \(n\) and \(k\)

Since \(m=n\cdot k\) and \((a-b) = m\cdot q\), we can write \((a-b) = n\cdot k\cdot q\), which shows that \((a-b)\) is a multiple of \(n\).

Proving the congruence

Since \((a-b)\) is a multiple of \(n\), we can say that \(n\mid (a-b)\). Therefore, according to the definition of modular congruence, we can conclude that \(a \equiv b(\bmod n)\). Thus, we have shown that if \(a, b, m\) and \(n\) are integers such that \(m\) and \(n\) are positive, \(n \mid m\) and \(a \equiv b(\bmod m)\), then \(a \equiv b(\bmod n)\).

Key concepts

Integer Divisibility

Integer divisibility is a fundamental concept in number theory. It revolves around the idea of one integer "dividing" another integer without leaving a remainder. If you ever encounter a statement like "n divides m," it is denoted mathematically as \( n \mid m \). This means that when \( m \) is divided by \( n \), the result is a whole number with no remainder.
Understanding this is crucial when dealing with problems in modular arithmetic. For example, if \( n \mid m \), it implies \( m = n \cdot k \) for some integer \( k \). This relationship helps us in breaking down expressions and understanding how one number is composed.
In the exercise, the assertion that \( n \mid m \) establishes a groundwork on which we prove further properties related to modular congruence. It essentially provides a stepping stone to deduce subsequent results.

Congruence Relation

The congruence relation in modular arithmetic is an extension of divisibility. It's a special way of saying two numbers leave the same remainder when divided by a certain number. In mathematical terms, two integers \( a \) and \( b \) are congruent modulo \( m \), represented as \( a \equiv b(\bmod \, m) \), if \( m \mid (a - b) \), meaning \( m \) divides the difference between \( a \) and \( b \).
This concept is essential as it helps us understand how different numbers can behave similarly with respect to division. For example, if we know \( a \equiv b(\bmod \, m) \) and that \( n \mid m \), we can infer new congruences, since \( (a - b) \) being a multiple of \( m \) makes it automatically a multiple of \( n \), which divides \( m \).
The congruence relation supports problem-solving skills in dealing with modular systems, enabling us to shift between moduli as seen in the exercise.

Number Theory

Number theory is a major branch of mathematics with a focus on the properties and relationships of integers. It explores phenomena such as divisibility, prime numbers, and their intriguing patterns. Often regarded as the "queen of mathematics," some concepts date back thousands of years and continue to captivate mathematicians today.
Within the realm of number theory, modular arithmetic is a powerful tool, allowing mathematicians to model and solve problems related to remainders. It simplifies complex arithmetic operations into cycles and patterns, which helps in understanding problems involving divisibility and congruence.

• Integer divisibility ensures logical decomposition of numbers.
• Congruence relations streamline the way we interpret arithmetic operations.
• Number theory brings these concepts together, providing a robust framework for solving problems.

In the given exercise, we saw a practical application of these number-theoretical concepts, highlighting the elegant simplicity modular arithmetic can offer in transforming and solving integer problems.