On a highway around the outskirts of a city are a bridge, a tunnel and a dam in clockwise order, and only clockwise traffic is allowed. There is a toll charge of 5 to drive over the bridge, 13 to drive through the tunnel and 8 to drive on the dam. Miss Dos lives and works along the highway, driving to work in the morning and driving home in the evening.
One day at the office, she calculates that her total toll charge since she started working is 2101.
Counting in clockwise order, where are her home and office, between the bridge and the tunnel, between the tunnel and the dam, or between the dam and the bridge?
-->---B,5--->--T,13--->---D,8--->---
As calculation is done at office, so there is an extra trip to office. Let there be $n$ days to office, hence $(n+1)*x + n*y= 2101$.
An easy way can be to subtract $x$ from $2101$, and find out by substituting all values of $x$, which can be the fit value.
$x \in \{8, 21, 26,0\}$, with the corresponding value of $y = \{18, 5, 0,26\}$.
1. $x=8\implies (x+y=)26 \nmid 2093$,
2. $x=21\implies 26 \mid 2080$,
3. $x=26\implies 26 \nmid 2075$,
4. $x=0\implies 26 \nmid 2101$
So, home is between Bridge & Tunnel, while office is between Dam and Bridge.
Want to know an alternate way, more compact and may be advanced too.
A more compact way is to observe that the remainder of $2101$ on division by $26$ is $21,$ so she paid $21$ to get to the office today. This is often written $2101 \equiv 21 \pmod{26}.$