Problem 19 - A Computational Introduction to Number Theory and Algebra

Question

Let \(n\) be an integer. Generalizing Exercise \(1.11,\) show that if \(\left\\{a_{i}\right\\}_{i=1}^{k}\) is a pairwise relatively prime family of integers, where each \(a_{i}\) divides \(n\), then their product \(\prod_{i=1}^{k} a_{i}\) also divides \(n\).

Step-by-Step Solution

Answer

Question: Show that if a family of pairwise relatively prime integers divides an integer n, then the product of all the integers in the family also divides n. Answer: By applying the property that if a and b are relatively prime, and both a and b divide an integer n, then their product ab also divides n. Using induction, we have shown that this property holds for any family of pairwise relatively prime integers, where each integer divides n. Hence, the product of all the integers in the family also divides n.

Step 1: Write the property for a pair of relatively prime integers

If \(a\) and \(b\) are relatively prime (meaning the gcd\((a, b) = 1\)), and both \(a\) and \(b\) divide an integer \(n\), then their product, \(ab\), also divides \(n\).

Step 2: Show the property holds for the entire family

We will now show that the property holds for the entire pairwise relatively prime family of integers \(\left\\{a_{i}\right\\}_{i=1}^{k}\). Since all integers in the family are pairwise relatively prime, we know that for any distinct integers \(a_{i}\) and \(a_{j}\), \(1 = \text{gcd}(a_i, a_j)\). We will prove this property by induction.

Step 3: Induction base case

For the base case, consider the case when \(k = 2\). We have a family of two relatively prime integers, \(\left\\{a_1, a_2\right\\}\), where each integer divides \(n\). From step 1, we know that the product \(a_1 a_2\) divides \(n\) when \(a_1\) and \(a_2\) are relatively prime. Thus, the induction base case holds.

Step 4: Induction step

Assume that for any family of \(k\) pairwise relatively prime integers, their product divides \(n\). Now, consider a family of \(k + 1\) pairwise relatively prime integers \(\left\\{a_i\right\\}_{i=1}^{k+1}\). Let \(b = \prod_{i=1}^{k} a_i\). Since, by the induction assumption, \(b\) divides \(n\), and \(a_{k+1}\) also divides \(n\), we have a new family of two relatively prime integers \(\{b, a_{k+1}\}\) that divides \(n\). From step 1, we know that the product of relatively prime integers \(b\) and \(a_{k+1}\) also divides \(n\). Therefore, we get the product \(\prod_{i=1}^{k+1} a_i\) divides \(n\). Thus, the induction step holds.

Step 5: Conclusion

By the principle of mathematical induction, we have shown that for any family of pairwise relatively prime integers \(\left\\{a_i\right\\}_{i=1}^{k}\), where each integer divides \(n\), the product \(\prod_{i=1}^{k} a_i\) also divides \(n\).

Key Concepts

Relatively Prime Integers

Understanding relatively prime integers is crucial in number theory. Two integers are considered relatively prime if their greatest common divisor (GCD) is one. In simpler terms, they do not share any common positive factors other than 1.
Being relatively prime does not mean that both numbers are prime, it just means that they do not divide each other except for the trivial case of one dividing both. For example, 8 and 15 are relatively prime since their only common divisor is 1.
Relating this concept to the problem, if you have a set of integers that are pairwise relatively prime, each pair in this set has a GCD of 1. This property helps maintain divisibility when combined, as seen in the exercise where each integer in the set divides another integer, and their product collectively divides the integer.

Mathematical Induction

Mathematical induction is a powerful proof technique used extensively in mathematics to establish statements or properties about an infinite set. Primarily, it allows you to prove that a statement is true for all natural numbers by applying it in two critical steps.

• Base Case: Verify that the statement holds for a starting point, often the smallest integer in the domain.
• Inductive Step: Assume the statement is true for some arbitrary integer \(k\), and then prove it holds for \(k + 1\).

In the context of the problem, induction is applied to prove that if each number in a family of pairwise relatively prime integers divides a number, then the entire product of these integers also divides the number. We start by checking if the product of any two relatively prime integers divides the number. Then, assuming the statement true for \(k\) integers, we extend it to \(k + 1\) integers, thereby completing the proof using induction.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) of two integers is the largest positive integer that divides both numbers without leaving a remainder. Essentially, it is a way to determine the largest factor shared by two numbers.
For example, the GCD of 12 and 18 is 6, because 6 is the largest integer that can evenly divide both numbers. The GCD can be determined using different methods like the Euclidean algorithm.
In the scenario of pairwise relatively prime integers described in the exercise, the GCD is particularly important as it equals 1 for every distinct pair. This property ensures that the product of such integers can divide another integer without any remainder, provided each also individually divides the number. The concept of GCD ties back to the idea of relatively prime numbers and plays a key role in solving the given problem through mathematical induction.