Chapter 10: Problem 3

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

Short Answer

Expert verified

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 \( a = b \)'.

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

Short Answer

Step by step solution

Set up the problem

Prove forward implication

Prove reverse implication

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