In the US you can donate whole blood every eight weeks. Suppose Elvira does exactly that. Will she hit every date of the year? For example, if Elvira gave blood today, will she in some future year give blood on the 4th of July? Can we figure it out without having to rely on a computer simulation or even a calculator?
Let's make the assumptions that the blood center is open every day and that Elvira gives blood exactly every 56 days for eternity.
A year has 365 days which is relatively prime to 56=23*7 since 365 mod 2 =1 and 365 mod 7 = 1. By the Chinese remainder theorem her next 365 blood donations will be on 365 distinct dates. If Elvira started giving blood at age 17, she will have hit every date at age 73.
That was easy but wrong. We have to account for those pesky leap years.
In a four year span, there will be one leap day. The total days in four years (using modular arithmetic) will still be odd and 5 mod 7, so still relatively prime to 56. So Elvira will donate on every day on the calendar exactly four times, except February 29th which she will hit once, over a period 56*4=224 years.
Alas not quite. Years ending 00 are not leap years, unless the year is divisible by 400. 2000 was a leap year but 2100 won't be. Any stretch of 224 years will hit at least one of those 00 non-leap years.
In 400 years, there will be 97 leap years. Since a regular year is 1 mod 7 days and a leap year is 2 mod 7 days, 400 years will be 497 mod 7 days. Since 497=71*7, 400 years has a multiple of 7 days. Every 400 years we have exactly the same calendar. February 3, 2421 is also a Wednesday.
The cycle of blood donations will repeat every 3200 years, the number of years in the least common multiple of 56 and the odd multiple of seven number of days in 400 years. But we can no longer directly apply the Chinese remainder theorem and argue that every day of the year will be hit. In those 3200 years Elvira will have over 20,000 blood donations. If the dates were chosen randomly the expected number to hit all dates would be 2372 by coupon collector. So one would expect Elvira would hit every day, but that's not a proof.
So I had to dust off my Python skills and do the computer simulation after all. No matter what day Elvira starts donating she will eventually hit every date of the year. If Elvira starts donating today, she would give blood on the 4th of July for the first time in 2035 and hit all dates on January 8, 2087 after 431 donations. The longest sequence is 3235 donations starting April 25, 2140 and hitting all dates on February 29, 2636.