Question

The set of real values of $x$ satisfying the equality $\left[\frac{3}{x}\right]+\left[\frac{4}{x}\right]=5$ (where $[.]$ denotes the greatest integer function) belongs to the interval $(a,\frac{b}{c}]$ where $a,b,c$ $\in$ $N$ and $\frac{b}{c}$ is in its lowest form. Find the value of $a+b+c+abc$.

• A. $25$
• B. $18$
• C. $20$
• D. $10$

Answer 1

The correct option is C $20$

Given, $\left[\frac{3}{x}\right]+\left[\frac{4}{x}\right]=5,$

$2\le \frac{3}{x}<3,3\le \frac{4}{x}<4,$

$x\le \frac{3}{2},x>1andx\le \frac{4}{3},x>1$

$i.e,x\le \frac{4}{3},x>1⟹xϵ(1,\frac{4}{3}]$

$\therefore a=1,b=4,c=3$

$a+b+c+abc=1+4+3+\mathrm{1.3.4}=20$