Problem 15 - Algebra 1: Concepts and Skills

Question

Find a counterexample to show that the statement is not true. If \(a\) and \(b\) are integers, then \(a \div b\) is an integer.

Step-by-Step Solution

Answer

A counterexample is provided by setting \(a = 1\) and \(b = 2\). The division result in this case is 0.5, which is not an integer. Therefore, the given statement is false.

Step 1: Understanding the statement

The given statement is: 'If \(a\) and \(b\) are integers, then \(a \div b\) is an integer.' This states that if you divide any integer \(a\) by any integer \(b\), the result would always be another integer. Remember an integer is a whole number, it can be either positive, negative or zero.

Step 2: Thinking of possible counterexamples

In order to disprove the statement, it is necessary to think about possible integer combinations where the division does not result in an integer. If \(a\) and \(b\) are integers, then a key place to start might be when \(b > a\) and \(b \neq 1\) as this could lead to a fraction, which is not an integer.

Step 3: Providing specific counterexample

A specific example could be: Let \(a = 1\) and \(b = 2\). Therefore, \(a \div b = 1 \div 2 = 0.5\). The result 0.5 is not an integer, which invalidates the original statement. Hence, the statement 'If \(a\) and \(b\) are integers, then \(a \div b\) is an integer.' is not always true.