I am looking over some of the british mathematical olympiad questions and tried the following question:
On Thursday 1st January 2015, Anna buys one book and one shelf. For the next two years, she buys one book every day and one shelf on alternate Thursdays, so she next buys a shelf on 15th January 2015.
On how many days in the period Thursday 1st January 2015 until (and including) Saturday 31st December 2016 is it possible for Anna to put all her books on all her shelves, so that there is an equal number of books on each shelf?
I have come to the conclusion that the only time that Anna has enough books such that there are an equal number of books on each shelf, is on an odd week, except for the second week.
Since there are $104$ whole weeks, $52$ of them will be odd. Adding $1$ due to the second week working we get 53. Hence the number of days there is an equal number of books on each shelf is $53 \times 7 = 371$ days.
Is my reasoning and answer correct, or not?