Question

If \(a \in \mathbb{Q}\) then \(5 a \in \mathbb{Q},\) and if \(5 a \in \mathbb{Q}\) then \(a \in \mathbb{Q}\)

Short answer

The two statements are true since they follow directly from the definition of rational numbers

Step-by-step solution

Demonstration of the first statement

For the first statement, we start with the assumption that \(a \in \mathbb{Q}\), this means \(a\) is a rational number. Rational numbers are numbers that can be expressed as a quotient of two integers; therefore \(a = \frac{p}{q}\), where \(p, q \in \mathbb{Z}\), and \(q \neq 0\). If we multiply \(a\) by 5, the operation is distributed to the numerator \(p\) of the fraction, this results to \(5a = 5\frac{p}{q} = \frac{5p}{q}\). This expression maintains the quotient format of two integers which is the definition of a rational number. Thus, if \(a \in \mathbb{Q}\) then \(5a \in \mathbb{Q}\).

Demonstration of the second statement

For the second part, let’s start with the given that \(5a \in \mathbb{Q}\), meaning 5a is a rational number. Therefore, we can express it as \(\frac{p}{q}\), where \(p, q \in \mathbb{Z}\), and \(q \neq 0\). After dividing both sides of \(5a = \frac{p}{q}\) by 5, we obtain the expression \(a = \frac{p}{5q}\), which is still a quotient of two integers. This shows that \(a\) is also a rational number. Therefore, we can conclude that if \(5a \in \mathbb{Q}\) then \(a \in \mathbb{Q}\).

Concluding the Demonstration

Both the first and second statements have been demonstrated to be true. The first statement proves that the product of a rational number and any integer (in this case 5) will result in a rational number. The second statement proves that any rational number divided by an integer (in this case 5) will also result in a rational number. These are fundamental properties of rational numbers.