How do I know if the number is divisible by $60$? I was study Babel Civilization ( Babylon ) , I see what about the number $60$ now I'm going to find a trick or fast method to know the rest of divisibility by number $60$ for example : 
$6689=60^{2}+51.60+29$ 
$2567=42.60+47$ 
I know that the number $60$ divisible by $1,2,3,4,5,6$ 
So I need fast know method I mean how the rest ? 
I don't if we can generalized or no ? 
Please I need fast method without using calculator !
For example $17894=?$ 
See that : $2177,12=2177+0,12$ 
$2177=36.60+17$
$0,12=\frac{3}{25}=\frac{60.3}{25.60}=?$ 
I need written in base $60$ ?

Here a method to find the rest of division of $N$ by $60$
1)Finding the rest for $10^n$ we put $t=40$ and $x=60$
$$10^2=x+t\\10^3=16x+t\\10^4=2x^2+46x+t\\10^5=27x^2+46x+t\\10^6=4x^3+37x^2+46x+t\\......\\......\\10^n=Mx+t\space  \text{for }n\ge3 $$ It follows a method we apply here particularly to the number $17894$ proposed by the O.P.
$$17894=Nx+(1+7+8)t+94=Nx+16\cdot40+94=Nx+734$$ it is easy to verify now that $17894=Nx+734\equiv734\equiv14\pmod{60}$
Method.-If $N=a_na_{n-1}\cdots a_2a_1a_0\equiv x\pmod{60}$ then $x\equiv 40(a_n+\cdots a_2)+a_1a_0$ so if $a_n+\cdots +a_2\equiv h\pmod3$ then $$\boxed{x\equiv 40h+a_1a_0\pmod{60}}$$ where $h$ is equal to $0,1$ or $2$.
