Question

Show that if \(a_{1}, a_{2}, \ldots, a_{n}\) are integers that are not all 0 and \(c\) is a positive integer, then \(\left(c a_{1}, c a_{2}, \ldots, c a_{n}\right)=c\left(a_{1}, a_{2}, \ldots a_{n}\right)\)

Short answer

Question: Show that the greatest common divisor (GCD) of \(c\) times the integers \(a_{1}, a_{2}, \ldots, a_{n}\) is equal to \(c\) times the GCD of the integers themselves. Answer: To show this, we first let \(d_1\) be the GCD of the given integers and \(d_2\) be the GCD of the integers after multiplying each by \(c\). Our goal is to show that \(d_2 = cd_1\). We rewrite the integers as a multiple of their GCD and substitute the expression for \(a_i\) into the expression for \(ca_i\). We then show that \(d_1\) divides both sides of the equation and derive the expression for \(cd_1\). Finally, we manipulate the equation to establish the desired relationship between \(d_1\) and \(d_2\), proving that the GCD of \(c\) times the integers is equal to \(c\) times the GCD of the integers themselves.

Step-by-step solution

Establish a general expression for the greatest common divisor of each case.

Let \(d_1 = (a_1, a_2, \ldots, a_n)\) be the GCD of the given integers, and let \(d_2 = (ca_1, ca_2, \ldots, ca_n)\) be the GCD of the integers after multiplying each by \(c\). Our goal is to show that \(d_2 = cd_1\).

Use the property of the greatest common divisor to rewrite the integers as a multiple of their GCD.

For every integer \(a_i\), we can write it as \(a_i = d_1x_i\), where \(x_i\) is an integer, since \(d_1\) divides each integer in the list. Likewise, we can write down every \(ca_i\) as \(ca_i = d_2y_i\), where \(y_i\) is another integer.

Substitute the expression for \(a_i\) into the expression for \(ca_i\).

With the expressions for \(a_i\) and \(ca_i\) established in step 2, substitute \(a_i\) from the expression for \(d_1\) into the expression for \(d_2\): \(ca_i = cd_1x_i = d_2y_i\)

Show that \(d_1\) divides both sides of the equation and derive the expression for \(cd_1\).

Since \(d_1\) divides each \(a_i\), it must also divide \(cd_1x_i\). Also, as \(d_2\) is the GCD of all the integers \(ca_1, ca_2, \ldots, ca_n\), it is also a divisor of the integer combination \(ca_i\). Thus, \(d_1\) should divide the right side of the equation as well: \(d_1 | d_2y_i\) Now we divide both sides of the equation by \(d_1\): \(c =\dfrac{d_2y_i}{d_1}\)

Use the divisibility property and manipulate the equation to establish the desired relationship between \(d_1\) and \(d_2\).

Since \(d_1 | d_2y_i\), we can express it in the form \(d_2y_i = d_1k\) for some integer \(k\). Substitute this expression into the previous equation: \(c = \dfrac{d_1k}{d_1}\) Now we divide by \(d_1\) to get \(c=k\). Now substitute this result back into \(d_2y_i = d_1k\): \(d_2y_i = d_1c\) Since this equation holds for every integer \(i\), we have proved that \(d_2 = (ca_1, ca_2, \ldots, ca_n) = c(a_1, a_2, \ldots, a_n) = cd_1\).

Key concepts

Integer Multiplication

Understanding integer multiplication is crucial for solving many mathematical problems, including those related to the greatest common divisor (GCD). In this exercise, we explored how multiplying each integer of a set by the same positive integer impacts the overall GCD of the set.
Multiplying each integer by a constant does not alter the relative divisibility among them. It simply scales each number by the same factor. This means that their common divisors will also be multiplied by this constant factor, transforming the GCD according to this scale.
Let's break this down: If you have a set of integers such as \(a_1, a_2, \ldots, a_n\) and multiply each by \(c\), the operation becomes \(ca_1, ca_2, \ldots, ca_n\). Though the values are now larger, the relationship between them remains unchanged, allowing us to find a direct connection between their GCD and the GCD of the original integers. This underpins the crux of our proof.

Divisibility

Divisibility is a central concept when discussing the greatest common divisor. Divisibility means that for integers \(a\) and \(b\), \(a\) is divisible by \(b\) if there exists an integer \(k\) such that \(a = bk\).
In the exercise, understanding divisibility allows us to express each integer as a product of its GCD and another integer, simplifying our calculations significantly. For instance, if \(d_1\) is the GCD of \(a_1, a_2, \ldots, a_n\), then each \(a_i\) can be expressed as \(a_i = d_1x_i\), where \(x_i\) is an integer.

• This expression helps us determine the impact of multiplying each integer by \(c\).
• By maintaining divisibility relationships, we confirm that scaling by \(c\) does not disrupt existing dependencies but simply transforms them.

Recognizing these relationships is key to establishing how the GCD is affected in the scaled set, and helps in proving that \((ca_1, ca_2, \ldots, ca_n) = c(a_1, a_2, \ldots, a_n)\).

Integer Expression

In mathematics, expressing numbers in terms of their divisibility factors is a powerful technique. It helps transform complex problems into simpler, more manageable forms.
In our exercise, we took each integer \(a_i\) and expressed it in terms of \(d_1\), its GCD with the others in the set. Specifically, we used the expression \(a_i = d_1x_i\), where \(x_i\) belongs to the set of integers. This transformation is especially helpful when scaling integers by a factor \(c\).

• The scaled integer becomes \(ca_i = cd_1x_i\).
• For the new list involving \(ca_1, ca_2, \ldots, ca_n\), we expressed its GCD as another product: \(d_2 = d_1 \cdot c\).

Creating these expressions allowed us to link the new GCD, \(d_2\), back to \(d_1\) through multiplication by \(c\). This clarity in expression is a cornerstone of mathematical proof and facilitates our understanding of the core relationship.

Mathematical Proof

A mathematical proof is a logical argument demonstrating the truth of a statement beyond doubt. Our goal was to prove that the GCD of a scaled set of integers is simply the original GCD multiplied by the scaling factor. This proof relies on logical reasoning through established properties and operations like integer multiplication and divisibility.
The proof begins by expressing both the original and scaled integers in terms of their respective GCDs. By manipulating these expressions, we demonstrate that dividing both sides by \(d_1\) shows that \(c\) remains constant and crucially linked to the new GCD.

• We concluded the proof by demonstrating that if \(d_2y_i = d_1c\) holds for every integer involved, then \(d_2 = cd_1\).

This completion of the proof rests on utilizing properties of numbers systematically. By showing that the scaled product maintains its divisibility relationships, we affirm the innate stability of the GCD under scaling transformations. This logical chain validates our original statement about \((ca_1, ca_2, \ldots, ca_n) = c(a_1, a_2, \ldots, a_n)\), providing a solid foundation for further exploration in numerical patterns and algebraic structures.