Counting the Number of Common Multiples Under $1000$ Liz and Sara start new jobs on the same day. Liz works three days in a row followed by $1$ rest day. Sara works $7$ days in a row followed by $3$ rest days. How many days between Day $1$ and Day $1000$ will they both have a rest day ? I know the answer is $100$, but how does this come about ?
 A: We could answer this with number theory, but this particular
problem it is easy enough to solve without invoking a lot of equations.
First, write down the first $20$ days of the combined schedule showing which days are rest days for Liz, which are rest days for Sara, and which are rest days for both.
Now look at the next $20$ days of the combined schedule. 
Can you detect a repeating pattern?
Once you spot the pattern, count the number of shared rest days in 
each repetition of the pattern
and figure out how many times the pattern repeats in $1000$ days.

Note that some problems like this will have only a partial repetition of the pattern at the end, because the length of the repeating pattern does not
exactly divide the length of the entire time period.
Does that happen in this problem?

Once you get the idea of how such a pattern works, you might think about how you could tell the length of the pattern just by looking at the numbers stated in the problem. And then consider how a little number theory might answer the question even if the pattern were longer than you would like to write out by hand.
A: Day $x$ is a rest day for Liz if $x=0\pmod4$.
Day $x$ is a rest day for Sara if $x=8 \text{ or } 9\text{ or } 0\pmod{10}$.
So you have to solve $3$ systems of congruence with the chinese remainder theorem.  
A: Well... I see this problem in the following way:
Observation $1$
Liz is working for three consecutive days and taking rest in fourth day, so the rest day for Liz occurs on multiples of all the multiples of $4$.
Observation $2$
Sara work for seven consecutive days and take a rest of three days. So, let's call one batch equals ten days and first seven days of the batch are work days while last three days are rest days.
There will be $1000/10=100$ batches in $1000$ days. 
Notice that of we are dealing with odd batch ($1st$, $3rd$ etc.. ) then we will have first rest day on the day which is multiple of $4$ ( $8,28,48..$) and while dealing with even batch, we will have last rest day as a multiple of $4$. Since there are only three consecutive rest days so we can't have two rest days in multiple of $4$ in the same batch.
Hence Sara will have only $100$ rest days as multiple of four. 
The above two observations make it clear that the number of days in which they will be sharing rest day are $100.$
