Question

Find the real integral solution of $\left[x\right]+\left[5x\right]+\left[10x\right]+\left[20x\right]=36K+35$, k$ϵ$integer where [] denotes the greatest integer function.

Answer 1

Since $\left[x\right]$ stands for greatest integer function, let us assume, $\left[x\right]=x$, that is x is an integer.
Then
$\left[x\right]+\left[5x\right]+\left[10x\right]+\left[20x\right]=36k+35$
$x+5x+10x+20x=36k+35$
$36x=36k+35$
$x=k+\frac{35}{36}$
Now $\frac{35}{36}$ is not an integer.
Hence
$x=k+\frac{35}{36}$ is not an integer.
This contradicts our initial assumption.
Hence not solution is possible.