Problem 27

Question

Without using the fact that every integer has a unique factorization into primes, prove that if $\operatorname{gcd}(a, b)=1$ and if $a \mid b c$, then $a \mid c$.

Step-by-Step Solution

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Answer

If $a \mid bc$ and $\operatorname{gcd}(a, b) = 1$, then $a \mid c$ by using Bezout's lemma and properties of division.

1Step 1: Understanding the Problem Statement

We need to show that if two integers $a$ and $b$ are coprime, i.e., $\operatorname{gcd}(a, b) = 1$, and $a$ divides the product $bc$, then it must also divide $c$. This must be shown without assuming the unique factorization theorem.

2Step 2: Expressing Divisibility

Since $a \mid bc$, there exists an integer $k$ such that $bc = ak$. Our goal is to write $c = aq + x$ for some integers $q$ and $x$ such that $a$ divides $c$.

3Step 3: Using the Coprimality Condition

Given $\operatorname{gcd}(a, b) = 1$, by Bezout's lemma, there exist integers $x$ and $y$ such that $ax + by = 1$. We can use these integers to help express $c$ in terms of $a$.

4Step 4: Expressing Multiples in Terms of Variables

Multiply the equation from Step 3 by $c$ to obtain: \[ c(ax + by) = c. \] This simplifies to: \[ acx + bcy = c. \] We already know $bc = ak$, so substitute in: \[ acx + aky = c, \] which further simplifies to \[ a(cx + ky) = c. \]

5Step 5: Conclusion from the Expression

Since $a(cx + ky) = c$, this expression shows that $a$ times some integer ($cx + ky$) equals $c$. This confirms that $a$ divides $c$ because it is expressed as a product involving $a$.

Key Concepts

Coprime IntegersBezout's LemmaDivisibility

Coprime Integers

Coprime integers are two numbers that have no common positive divisor other than 1. This means that the greatest common divisor, or GCD, of these numbers is 1. When two numbers are coprime, they are relatively prime to each other. For example, 8 and 15 are coprime.

There are no common factors except 1.
Their GCD is always 1.
They can share multiples, but not both be factors of any number other than 1.

Understanding coprime integers is key in number theory. It often provides insights into problems involving divisibility and integer solutions. Knowing that two numbers are coprime simplifies many mathematical proofs, since many properties of numbers become easier to handle without shared factors.

Bezout's Lemma

Bezout's Lemma is a fundamental statement in number theory. It deals with linear combinations of integers. According to Bezout's Lemma, if two integers, say $a$ and $b$, are coprime (i.e., their GCD is 1), then there exist integers $x$ and $y$ such that $ax + by = 1$. This is a powerful tool because it tells us that using combinations of $a$ and $b$, we can generate any integer that is a multiple of their GCD, which is 1 in the case of coprime numbers.

It provides a way to express 1 as a linear combination of $a$ and $b$.
This expression is essential in solving many problems where divisibility comes into play.
Bezout's Lemma is also useful in proving the existence of integer solutions to certain equations.

In the step-by-step solution, using Bezout's Lemma allows us to manipulate the terms and establish the divisibility condition required by the problem. It connects the abstract concept of coprimeness with the concrete ability to express integers as combinations, thereby facilitating further solutions.

Divisibility

Divisibility refers to the ability of one integer to be divided by another integer without any remainder. Formally, an integer $a$ divides another integer $b$ (denoted as $a \mid b$) if there exists an integer $k$ such that $b = ak$. This is a central concept in number theory and underlies many deeper properties and theorems.

Divisibility helps classify numbers in terms of their factors.
It lays the groundwork for theorems like Bezout's and properties of prime numbers.
Understanding divisibility is crucial for tackling problems in modular arithmetic and other areas.

In the given exercise, divisibility plays a critical role. The proof involves using the fact that $a$, which divides $bc$, can be related back to $c$ itself, under specific conditions derived from the concept of coprime integers and Bezout's Lemma. This shows us how interconnected these foundational topics in number theory are, each reinforcing the others.

Problem 26

Problem 30

Other exercises in this chapter

Problem 23

(a) Find a single value $x$ that simultaneously solves the two congruences $$ x \equiv 3 \quad(\bmod 7) \quad \text { and } \quad x \equiv 4 \quad(\bmod 9) $$

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Problem 26

Let $\left\\{p_{1}, p_{2}, \ldots, p_{r}\right\\}$ be a set of prime numbers, and let $$ N=p_{1} p_{2} \cdots p_{r}+1 . $$ Prove that $N$ is divisible by so

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Problem 30

For each of the following primes $p$ and numbers $a$, compute $a^{-1} \bmod p$ in two ways: (i) Use the extended Euclidean algorithm. (ii) Use the fast po

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Problem 32

Recall that $g$ is called a primitive root modulo $p$ if the powers of $g$ give all nonzero elements of $\mathbb{F}_{p}$. (a) For which of the following

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