How can I find the day of the week which an event happened after some time has elapsed?

Question

The problem is as follows:

An astronomer discovers a new comet while scouting the skies on Monday but decides to report his discovery $256$ days after. What day of the week did the astronomer found the comet?

I'm lost at this problem. I'm not sure if this is related with divisibility. But I've found by following the rules of divisibility of $7$ since it is the number of days of the week and the number given $256$ is not divisible by $7$ as:

$$256:\,\textrm{is divisible if}\,25-2\times6=25-12=13$$

Therefore is not divisible by $7$, then by trying other numbers I've found that the closest lower boundary number $252$ is divisible by $7$. Since

$$252:\,\textrm{is divisible if}\,25-2\times2=25-4=21\,\textrm{Check!}$$

Then $256=252+4$ Then should four days of the week starting from monday as that day would repeat after $252$ days since a week has seven days and therefore additional four days would be need to be added:

$\textrm{monday + 4 days= saturday?}$

I'm not sure if the right answer would be that day or Friday or Saturday, will it account for Monday as one day?. Not sure if what I did was right can somebody put me on the right track?.

Answer 1

If astronomer made discovery on Monday and reported it $7$ days later, it would be Monday again. If he reported it one day later, it would be Tuesday. You found out that $256$ days is $36$ full weeks plus four days. Thus, the problem is equivalent to this: if discovery was made on Monday and reported four days later, what day of the week that would be? It looks like Friday to me.

Answer 2

Here is a way to generalize it:

Assume that an event $A$ happens on one day of the week (Monday, Tuesday,..., Sunday); an event $B$ happens $x$ days after the event $A$ happens.

There is exactly one number in $7$ numbers $x;x-1;x-2;x-3;x-4;x-5;x-6$ that is divisible by $7$ (because they are $7$ consecutive integers for all natural numbers $x\ge0$).

Assume that $x-k$ is divisible by $7$ ($k$ is either $0;1;2;3;4;5;6 $), put $x-k \div 7 = y$.

Then $x=7y+k$, which means event $B$ happens $7y+k$ days later, or $y$ weeks and $k$ days after event $A$ happens. Of course after $y$ weeks, the day of the week isn't changed, so we only have to put $k$ days ahead from the event $A$ to see which day of the week event $B$ happens.

For this problem, $x=256=36 \times 7 +4$, which means the comet is reported $36$ weeks and $4$ days later. Monday, Tuesday, Wednesday, Thursday, Friday, so the comet is reported on Friday.

To make you easy to understand, 4 days after tomorrow, March 1, 2018 will be March 5, 2018; not March 6, 2018. One day after March 1, 2018 will be March 2, 2018 and so on.