Question

Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters \(4-7\). Suppose \(a \in \mathbb{Z}\). Then \(a^{2} \mid a\) if and only if \(a \in\\{-1,0,1\\}\)

Short answer

The statement is proven to be true. If \(a^{2}\) divides \(a\), then \(a\) must be -1, 0 or 1. Conversely, if \(a\) is -1, 0, or 1, then \(a^{2}\) divides \(a\).

Step-by-step solution

Proof of the Forward Direction

Assume that \(a^{2}\) does indeed divide \(a\). This means that there is some integer \(k\) such that \(a = a^{2}k\). This can be rewritten as \(a(1-ak) = 0\). Therefore, either \(a = 0\) or \(1-ak = 0\). Solving the latter equation for \(a\) gives \(a = 1/k\). But, as \(a\) and \(k\) are both integers by definition, the only integer values of \(k\) for which \(a = 1/k\) would also be an integer are \(k = 1\) and \(k = -1\), giving \(a = 1\) and \(a = -1\) respectively. Therefore, if \(a^{2}\) divides \(a\), then \(a\) must be equal to -1, 0, or 1.

Proof of the Reverse Direction

Assume now that \(a\) is -1, 0, or 1. If \(a = 0\), then \(a^{2} = 0\) which divides \(a\). If \(a = 1\), then \(a^{2} = 1\) which divides \(a\). If \(a = -1\), then \(a^{2} = 1\) which also divides \(a\). Thus, in all cases, \(a^{2}\) divides \(a\).

Key concepts

Integer Division

Understanding the principle of integer division is essential in the realm of mathematics, especially when it comes to divisibility and algebra. Integer division relates to how one integer, called the dividend, can be divided by another, known as the divisor, resulting in a quotient that is itself an integer, without any remainder.

For example, if we divide 6 by 2, the quotient is 3; both the numbers are integers, hence it is a case of integer division. However, 7 divided by 2 equals 3.5, which is not an integer. In the context of integer division, the remainder must always be zero for it to be valid.

This concept is particularly relevant in the given exercise, where we are dealing with conditions where an integer squares itself and then divides itself. The integers that meet this condition without leaving a remainder are central to the solution of the exercise.

Divisibility

Divisibility is a fundamental concept wherein one integer can be divided by another with no remainder. It's a key building block not only in arithmetic but also in number theory. In formal terms, an integer \(a\) is divisible by another integer \(b\) if there exists an integer \(k\) such that \(a = bk\).

In practice, this means that when \(a\) is divided by \(b\), the result is an integer with no fractional part. It's essential to recognize certain divisibility rules, such as any even number is divisible by 2, and any number where the sum of its digits is divisible by 3 is itself divisible by 3.

The exercise provided highlights this concept by exploring cases where an integer's square is a divisor of the integer. The proof essentially revolves around the properties of divisibility to arrive at the conclusion concerning the values of the integer.

Proof Techniques

Proof techniques are the tools that mathematicians use to demonstrate the truth or falsity of mathematical statements. There are various methods of proof, including direct proof, proof by contradiction, and proof by induction. Each technique is best suited to different kinds of problems and understanding when and how to use them is crucial in mathematical reasoning.

In our exercise, we see a direct proof in action. This involves assuming that the statement we're trying to prove is true, then logically deducing consequences until reaching a conclusion that confirms the statement's validity. The proof given for the integer divisibility condition is broken down into two parts: proving that if \(a^2\) divides \(a\), then \(a\) must be -1, 0, or 1, and conversely, that if \(a\) is -1, 0, or 1, then \(a^2\) does divide \(a\).

Explicitly, the first part of the proof tackles the forward direction of the statement ('if') using direct proof, and the second part covers the reverse direction ('only if'), again using direct proof. This illustrates the inherent symmetry and beauty of mathematical logic.