Question

Given that a, b are two integers such that the positive integer solutions of the system of inequalities $9x - a \ge 0$, and $8x - b < 0$ are $1, 2, 3.$ Find the number of the ordered pairs $(a, b).$

• A. $72$
• B. $75$
• C. $81$
• D. None of these

Answer 1

Answer: (A)

The solution set of the system in the set of the integers $x$ satisfying
$\frac{a}{9} \le x< \frac{b}{8}.$ Since each of $1, 2, 3$ satisfies the two inequalities,
$0<\frac{a}{9} \le1 \Rightarrow 0 \le a\,9,$
$3 > \frac{b}{8} \le4 \Rightarrow 24 < b \le32.$
So a has 9 choices and b has 8 choices, i.e. there are $9 × 8 = 72$ required ordered pairs $(a, b).$