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

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

Tardigrade Admin · Accepted Answer

The solution set of the system in the set of the integers

x

satisfying

\frac{a}{9} \leq x < \frac{b}{8} .

Since each of

1, 2, 3

satisfies the two inequalities,

0 < \frac{a}{9} \leq 1 \Rightarrow 0 \leq a 9,

3 > \frac{b}{8} \leq 4 \Rightarrow 24 < b \leq 32.

So a has 9 choices and b has 8 choices, i.e. there are

9 \times 8 = 72

required ordered pairs

(a, b) .

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

$72$

$75$

$81$

None of these

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

72

75

81

None of these

The solution set of the system in the set of the integers x satisfying 9a​≤x<8b​. Since each of 1,2,3 satisfies the two inequalities, 0<9a​≤1⇒0≤a9, 3>8b​≤4⇒24<b≤32. So a has 9 choices and b has 8 choices, i.e. there are 9×8=72 required ordered pairs (a,b).

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

$72$

$75$

$81$