Show that days with the identical calendar date in the years 1999 and 1915 fell on the same day of the week.

Question

I think I'll be able to work this problem if I understand the question. I am having difficulty in interpreting the problem (the phrase "identical calendar date" is throwing me off). Any help is appreciated. Thanks!

Answer 1

As pointed out by Simon S in the comments, the question is asking you to show that January $1$ $1915$ and January $1$ $1999$ were both on the same day of the week. Likewise for January $2$, January $3$, $\dots$ , December $30$, and December $31$.

To do this, note that there are $84$ years between January $1$ $1915$ and January $1$ $1999$. Of these $84$ years, $21$ of them are leap years and $63$ are not. Therefore, the number of days between January $1$ $1915$ and January $1$ $1999$ is $D = 366\times 21 + 365\times 63$. Likewise, any date in $1999$ occurs $D$ days after the corresponding date in $1915$. As $21$ and $63$ are both divisible by $7$, $D$ is also divisible by $7$. As the days of the week repeat every seven days, they also repeat every $D$ days, so every date in $1915$ occurs on the same day as the corresponding date in $1999$.