Question

If \( A = \{ (a, b) : 4a = 5b, a \in \{1, 2, 3, \dots, 30\} \}, \) then the number of such ordered pairs \( (a, b) \) is:

• A. 4
• B. 6
• C. 8
• D. 10

Hint:

When given an equation relating two variables, solve for one variable and find the conditions for the other to be an integer or satisfy any restrictions.

Answer 1

The Correct Option is B

We are given the equation \( 4a = 5b \), and we are asked to find the number of ordered pairs \( (a, b) \) where \( a \in \{1, 2, 3, \dots, 30\} \) and \( b \) is determined by this equation. We can solve for \( b \) in terms of \( a \): \[ b = \frac{4a}{5} \] For \( b \) to be an integer, \( a \) must be a multiple of 5. Therefore, the possible values of \( a \) are \( a = 5, 10, 15, 20, 25, 30 \), and for each of these values of \( a \), we get a corresponding value of \( b \). Thus, there are 6 such pairs \( (a, b) \). Therefore, the correct answer is 6.