Solving $7[x]+23\{x\}=191$

For every real number $x$, $[x]$ denotes the largest integer less than or equal to $x$ and $\{x\}=x-[x]$.

The number of real solutions of

$$7[x]+23\{x\}=191$$ is

(a) 0 $\quad$ (b) 1 $\quad$ (c) 2 $\quad$ (d) 3

I solved it like this:

$$7[x]+23\{x\}=191$$

$$7(x-\{x\})+23\{x\}=191$$

$$7x+16\{x\}=191$$

Now,

$$0 \leq \{x\} < 1$$

$$0 \leq 16\{x\} < 16$$

$$0 \leq 191-7x < 1$$

$$-16 < 7x-191 \leq 0$$

$$175 < 7x \leq 191$$

$$\frac{175}{7} < x \leq \frac{191}{7}$$

$$25 < x \leq 27.2857$$

But $x=26$ doesn't satisfy given equation. Answer given is $(d)$ $3$.

I don't understand where I'm wrong.

A hint will be of great help.

Please don't provide complete solution.

Sorry for poor LaTeX.

Solution 1:

You should work with $[x]$ and not $x$, try $$0≤ \{x\} <1$$

$$0≤ 23\{x\} <23$$

$$0≤191−7[x]<23$$

$$−23<7[x]−191≤0$$

$$168<7[x]≤191$$

$$168/7<[x]≤191/7$$

$$24<[x]≤27.2857$$

So your answers are some $x$ with $[x]$ = $25$, $26$ or $27$

The $\{x\}$ part is just $\frac{(191 - 7[x])}{23}$

then :)

Solution 2:

Fractional part of x , $\{x\}$ must be of the form $\frac{y}{23}$. Thus, write $x = z + \frac{y}{23}$ where $z = [x]$ and $y<23$ and $y,z$ are integers. Thus, $7z + y = 191$. $$191 \equiv 2 \mod 7$$ Thus, $y \equiv 2 \mod 7$. Number of such values of $y$ that are less than 23 are just 3 (2,9,16).