Question

show that two integers divide each other if and only if they are equal.

Short answer

Proving the 'if' direction is straightforward: if two integers are equal, they obviously divide each other. For the 'only if' direction, if two integers divide each other, it's only possible when they are equal. If the integers were not equal, they wouldn't be divisors of each other. Therefore, we've proven that two integers divide each other if and only if they are equal.

Step-by-step solution

Set up the problem

Let \( a \) and \( b \) be integers. We need to prove that \( a \) divides \( b \) and \( b \) divides \( a \) if and only if \( a = b \). This is a biconditional statement, which means we need to prove two things: 'if \( a = b \) then \( a \) divides \( b \) and \( b \) divides \( a \)', and 'if \( a \) divides \( b \) and \( b \) divides \( a \) then \( a = b \)'.

Prove forward implication

If \( a = b \), then by definition \( a \) divides \( b \) because \( a = b \cdot 1 \), and \( b \) divides \( a \) because \( b = a \cdot 1 \). So the statement 'if \( a = b \) then \( a \) divides \( b \) and \( b \) divides \( a \)' is proven.

Prove reverse implication

Assume that \( a \) divides \( b \) and \( b \) divides \( a \). This means there exist integers \( m \) and \( n \) such that \( b = am \) and \( a = bn \). Substituting the first equation into the second we get \( a = a(mn) \). Division by \( a \) is permissible when \( a ≠ 0 \). Then the equation simplifies to \( 1 = mn \). The integers \( m \) and \( n \) can only equal either \(1\), \(-1\), meaning that \( a = ±b \). But since \( a \) and \( b \) are both divisors of each other, they have to be positive, leading to \( a = b \). This verifies the statement 'if \( a \) divides \( b \) and \( b \) divides \( a \) then \( a = b \)'.

Key concepts

Divisibility

Divisibility is a central concept in number theory. In simple terms, an integer \( a \) is said to divide another integer \( b \) if there exists an integer \( k \) such that \( b = a \cdot k \). This means \( b \) can be expressed as a multiple of \( a \), and there is no remainder left over.

• If \( a \) divides \( b \), we often write this relationship as \( a | b \).
• Divisibility is a foundational idea for understanding prime numbers, greatest common divisors, and least common multiples.
• If \( a = 1 \), then any integer is divisible by \( a \) because \( b = 1 \cdot b \).

Understanding divisibility helps solve many mathematical problems as it lays the groundwork for further exploration into more complex topics like modular arithmetic and factorization.

Integers

Integers are the set of whole numbers that include all positive and negative numbers, as well as zero. They form the backbone of number theory.

• Numbers like -3, 0, and 7 are all integers.
• Integers do not include fractions or decimals.
• They can be seen on the number line, extending infinitely in both positive and negative directions.

This makes them critically important in problems where whole quantities are essential—like counting, ordering, and equally significant in mathematical proofs.

Biconditional Statements

A biconditional statement is a logical statement that declares two conditions are logically equivalent—they hold true together or they do not hold true at all.

• Symbolically, this is represented as \( p \Leftrightarrow q \), meaning "\( p \) if and only if \( q \)".
• Such statements necessitate proving two implications: \( p \rightarrow q \) and \( q \rightarrow p \).
• They are useful in proofs where precision and clarity of relationships are necessary.

In the context of the exercise, the statement involves showing two integers are equivalent because one divides the other and vice versa.

Mathematical Proofs

Mathematical proofs are logical arguments that demonstrate the truth of a given statement or theorem. They are crucial for establishing certainty and rigor in mathematics.

• Proofs often begin with assumptions followed by logical deductions.
• They may involve direct proofs, contrapositive proofs, or contradiction.
• The purpose is to show that a statement holds true universally, not just in specific instances.

In number theory, proofs help formalize concepts such as divisibility and properties of integers. The exercise showcases an example of a biconditional proof, where two implications need to be established clearly, leveraging concepts such as divisibility.