Question

Use the method of direct proof to prove the following statements. Suppose \(a, b\) and \(c\) are integers. If \(a^{2} \mid b\) and \(b^{3} \mid c,\) then \(a^{6} \mid c\).

Short answer

If \(a^{2} \divides b\) and \(b^{3} \divides c\), then, indeed, \(a^{6} \divides c\) as it is shown through the method of direct proof.

Step-by-step solution

Understand the Given Information

We have three integers \(a\), \(b\), and \(c\). We are given that \(a^{2}\) divides \(b\) and that \(b^{3}\) divides \(c\). This means that \(b\) is an integer multiple of \(a^{2}\), and \(c\) is an integer multiple of \(b^{3}\). We are asked to prove that \(c\) is an integer multiple of \(a^{6}\). In other words, we need to prove that \(a^{6}\) divides \(c\).

Use the Given Information and Direct Proof

We know that \(a^{2} \mid b\), which means there exists some integer \(k\) such that \(b = k \cdot a^{2}\). Similarly, because \(b^{3} \mid c\), there exists some integer \(p\) such that \(c = p \cdot b^{3}\). Substituting the value of \(b\) from the first equation into the second, we get \(c = p \cdot (k \cdot a^{2})^{3} = p \cdot k^{3} \cdot a^{6}\). This shows that \(c\) is a multiple of \(a^{6}\), which proves that \(a^{6} \mid c\).

Conclusion

After utilizing the given information and following the steps of direct proof, we have proven that if \(a^{2} \divides b\) and \(b^{3} \divides c\), then \(a^{6} \divides c\).

Key concepts

Proof by Direct Method

In the realm of mathematics, the direct proof method is a straightforward way to confirm the validity of a statement. The process involves starting with known facts or accepted truths (axioms) and utilizing logical steps to show that another statement must be true as a result. This method can be powerful because it builds on the foundation of information that is already agreed upon to reach a new conclusion.

When it comes to the direct proof of statements involving divisibility, we follow the structure: assume the premises are true, and then, by using definitions and arithmetic, show that the conclusion necessarily follows. This approach is very systematic and is ideal for demonstrating properties like divisibility, where each step builds upon the previous one.

Divisibility in Integers

Divisibility in integers is a concept that focuses on dividing one whole number by another without leaving a remainder. An integer 'a' is divisible by another integer 'b' if there exists an integer 'k' such that the equation \(a = b \times k\) holds true. This relationship is critical, as it underpins many areas of number theory and arithmetic.

For instance, the statement \(a^2 \mid b\) means there's an integer 'k' that, when multiplied by \(a^2\), equals 'b'. Divisibility is a transitive property; if \(a\) divides \(b\), and \(b\) divides \(c\), then \(a\) divides \(c\) as well. This logical chain is heavily utilized in proving propositions within number theory using the direct proof method.

Mathematical Proof

A mathematical proof is a logical argument that verifies the truth of a mathematical statement beyond any doubt. It's a vital tool used to confirm that the conclusions drawn in mathematics are sound and reliable. Proofs come in various forms ranging from direct proofs to indirect methods like proof by contradiction or proof by induction.

In constructing a proof, each step must follow logically from the previous step. This rigorous logical foundation ensures the accuracy of the derived results and is essential for establishing theorems and other mathematical propositions. In the classroom, mastering mathematical proofs empowers students to approach complex problems systematically and with confidence.

Integer Multiples

An integer multiple of a number is obtained by multiplying the number by an integer. If 'n' is an integer, then any integer 'm' can be considered a multiple of 'n' if there exists an integer 'k' such that \(m = n \times k\). This simple yet fundamental concept is the cornerstone of understanding divisibility.

Integer multiples extend into various applications: they are used to define arithmetic sequences, determine common divisors and multiples, and help solve problems in modular arithmetic. Recognizing and working with integer multiples is an essential skill in mathematics, particularly when dealing with proofs concerning the divisibility and factors of numbers.