# Alternate solutions to an olympiad problem

I recently wrote the following question for a math olympiad:

Newton, one of the founding father's of Calculus passed away on the 31st of March $$1727$$. Lagrange, another one of the father's of Calculus was born on the $$25$$th of January $$1736$$. Given that: Sunday $$=0$$, Monday $$=1$$.... Saturday $$=6$$ and assuming that these dates follow the Gregorian calendar, determine the sum of the two days.

$$A)4$$ $$B)5$$ $$C)3$$ $$D)7$$ $$E)25$$

Firstly let us note that any date before the $$28$$th of February on a non leap year is the same date next year. Since there are $$365$$ days this implies the following congruence holds: $$365 \equiv 1$$ (mod $$7$$). The remainder of $$1$$ means that there is an increase of $$1$$ day in the week. So for example, if $$1$$st January is a Monday, it becomes Tuesday the following year.Using the previous logic, similar working shows that for a leap year, we $$+2$$ days.(e.g. "Monday" becomes "Wednesday".) Therefore:
Therefore if $$31/1/1732$$ was a Monday(which it actually was) then $$31/3/1735$$ is a Thursday. Following this logic and continuing it holds that $$25/1/1736$$ was a Wednesday. Since Wednesday $$= 3$$ and Thursday $$= 4$$, the sum is $$7$$, therefore the answer is $$D$$.