Question

Show that \(\operatorname{gcd}(a, b)=1\) and \(\operatorname{gcd}(a, c)=1\) imply that \(\operatorname{gcd}(a, b c)=1\).

Short answer

\(\operatorname{gcd}(a, bc)=1\) by the properties of coprimality and divisors.

Step-by-step solution

Understand the GCD Definition

The greatest common divisor (gcd) of two integers is the largest positive integer that divides both numbers without leaving a remainder. If \( \operatorname{gcd}(a, b) = 1 \), it means \(a\) and \(b\) are coprime, sharing no prime factors.

Use the Coprimality Properties

Given that \( \operatorname{gcd}(a, b) = 1 \) and \( \operatorname{gcd}(a, c) = 1 \), \(a\) has no common prime factors with either \(b\) or \(c\). Therefore, no factor of \(a\) divides either \(b\) or \(c\) except for 1.

Apply the GCD Property to the Product

The coprimality of \(a\) with both \(b\) and \(c\) implies \(a\) is also coprime with their product \(bc\). This is because if there were any common prime factor dividing \(a\) and \(bc\), it would also have to divide either \(b\) or \(c\), contradicting the coprimality conditions.

Conclude Based on the Logic

Since no common divisor greater than 1 exists for \(a\) with either \(b\) or \(c\), any divisor of \(a\) dividing \(bc\) must divide either \(b\) or \(c\). Therefore, \( \operatorname{gcd}(a, bc) = 1 \) follows from the individual coprimalities.

Key concepts

gcd (greatest common divisor)

The greatest common divisor, often abbreviated as gcd, is a fundamental concept in number theory. When we talk about the gcd of two integers, let's say \(a\) and \(b\), we are referring to the largest positive integer that can divide both \(a\) and \(b\) without leaving a remainder. For example, the gcd of 8 and 12 is 4 because 4 is the highest number that divides both without leftovers.

It’s important to highlight several key characteristics of the gcd:

• If \( ext{gcd}(a, b) = 1 \), then \(a\) and \(b\) are called coprime. It means they have no prime factors in common.
• The gcd function is both commutative and associative. That means \( ext{gcd}(a, b) = ext{gcd}(b, a) \) and \( ext{gcd}((a, b), c) = ext{gcd}(a, (b, c)) \).
• For any integer \(a\), \( ext{gcd}(a, 0) = |a| \).

Understanding these properties helps in solving problems related to common divisors, coprimality, and prime factorization.

coprime

When two numbers are coprime, their greatest common divisor is 1. In other words, coprime numbers do not share any prime factors other than 1. For example, 8 and 15 are coprime, as the only divisor they have in common is 1.

Being coprime is a useful property in number theory for several reasons:

• Coprime numbers often appear in problems involving fractions, where they need to be reduced to their simplest form.
• They are essential in modular arithmetic and cryptographic algorithms.
• If \(a\) is coprime with both \(b\) and \(c\), it implies \(a\) is coprime with the product \(b \cdot c\). This is because any factor of \(a\) dividing \(b \cdot c\) would necessarily divide \(b\) or \(c\), which contradicts the initial coprimacy assumption.

The concept connects closely with another pillar of number theory: prime numbers.

prime factors

Prime factors are the "building blocks" of numbers. Any number can be broken down into a product of prime numbers, which are prime factors. A prime number is a number greater than 1 that has no divisors other than 1 and itself.

Finding prime factors involves dividing the number by the smallest possible prime until you reach 1. For example, to find the prime factors of 28 we do the following steps:

• Start with 2 (the smallest prime): 28 ÷ 2 = 14.
• Continue with 2: 14 ÷ 2 = 7.
• The remainder, 7, is a prime itself. So, the prime factors of 28 are 2, 2, and 7.

The prime factorization is vital in finding gcds and understanding coprime relationships. When solving gcd problems, knowing the prime factors of each number can quickly identify common or non-common factors, hence establishing coprimacy or lack thereof.