Problem 4 Let $a, b,$ and $c$ be integ... [FREE SOLUTION]

91影视

Spiral Workbook for Discrete Mathematics

Harris Kwong

$Math Studyset 91影视 Explanations$ Math

2015 Edition

Chapter 5: Problem 4

Let $a, b,$ and $c$ be integers such that $a \neq 0 .$ Prove that if $a \mid b$ and $a \mid c,$ then $a \mid(s b+t c)$ for any integers $s$ and $t$.

Short Answer

Expert verified

Since $sb + tc = a(sk + tl)$, $a \mid (sb + tc)$.

Step by step solution

Understand Divisibility

Given that $a \mid b$, it means there exists an integer $k$ such that $b = ak$. Likewise, given $a \mid c$, there exists another integer $l$ such that $c = al$. Our goal is to use these relationships to prove divisibility of any linear combination $sb + tc$.

Express sb and tc in terms of a

Start by expressing $sb$ and $tc$ using the divisibility definitions from Step 1: $sb = s(ak) = a(sk)$ and $tc = t(al) = a(tl)$. Both $sb$ and $tc$ can be rewritten in terms of $a$.

Combine sb and tc

Combine the expressions: $sb + tc = a(sk) + a(tl)$. Factor $a$ out of the expression: $sb + tc = a(sk + tl)$. This shows that $sb + tc$ is a multiple of $a$.

Conclude the Proof

Since $sb + tc = a(sk + tl)$, it is clear that $a$ divides $sb + tc$ because $sk + tl$ is an integer. Therefore, $a \mid (sb + tc)$ for any integers $s$ and $t$. Our claim is thus proved.

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Key Concepts

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

Divisibility

Divisibility is a fundamental concept in discrete mathematics that deals with how one integer can be evenly divided by another, without leaving a remainder. When we say that an integer $a$ divides another integer $b$, denoted as $a \mid b$, it means $b$ can be expressed as $b = ak$, where $k$ is an integer. This implies that $b$ is a multiple of $a$. Understanding divisibility is crucial because it helps us to simplify expressions and solve equations more easily.

It allows us to determine relationships between numbers.
It's essential for proving properties involving integers, such as the sum or combination of multiples.
Divisibility plays a significant role in number theory and cryptography.

When we get into more complex expressions, knowing the basic principle of divisibility helps us dissect and understand underlying patterns and relationships in numbers.

Integers

Integers are the set of whole numbers that include positive numbers, negative numbers, and zero. This set is denoted as $\mathbb{Z}$, which stands for numbers like $..., -3, -2, -1, 0, 1, 2, 3, ...$. Integers are crucial in mathematics because they provide a solid foundation for various operations and functions:

They help in performing operations like addition, subtraction, and multiplication.
Integers allow us to understand positive and negative values, which are essential in real-world applications.
These numbers form the basis for more advanced topics in mathematics and computer science.

In the context of divisibility, integers are pivotal because they make it possible for numbers to be divided without worrying about fractions or decimals. For instance, if we are given that $a \mid b$ and $a \mid c$, we know there must be integers $k$ and $l$ such that $b = ak$ and $c = al$. This understanding is key to solving divisibility problems.

Linear Combination

A linear combination involves the summation of products of constants and variables. In mathematics, specifically in integer theory, a linear combination of two integers $b$ and $c$ can be expressed as $sb + tc$, where $s$ and $t$ are integers or coefficients.

This concept is used to create new expressions from existing ones by combining them in numerous ways.
It allows for flexibility in proving mathematical relationships, such as divisibility.
Linear combinations are integral in solving linear equations and systems.

When it comes to proving the divisibility statement $a \mid (sb + tc)$, understanding linear combinations becomes critical. By expressing $sb$ and $tc$ in terms of $a$, we can factor out $a$, thus showing that the whole expression $sb + tc$ is divisible by $a$. This is why linear combinations are a powerful tool in algebra and number theory.

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91影视

Let \(a, b,\) and \(c\) be integers such that \(a \neq 0 .\) Prove that if \(a \mid b\) and \(a \mid c,\) then \(a \mid(s b+t c)\) for any integers \(s\) and \(t\).

Short Answer

Step by step solution

Understand Divisibility

Express sb and tc in terms of a

Combine sb and tc

Conclude the Proof

Key Concepts

Divisibility

Integers

Linear Combination

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