How many odd days are there in 1600 years ? I calculated it but I guess the result is wrong?

Question

I had to calculate the number of odd days in 1600 years. I have read the answer to be equal to 0. But I don't get it to equal to 0.

This is the way I am calculating the number of odd days in 1600 years :

1600 years = 24 x 16 = 384 leap years   (100 years = 24 leap years)
(because 100 years have 24 leap years)
1 leap year = 2 odd days (52 weeks + 2 odd days)
384 leap years = 384 x 2 = 768 odd days --(A)
1600 years = 1600 - 384 = 1216 ordinary years
1 ordinary year = 1 odd day (52 weeks + 1 odd day)
1216 ordinary years = 1216 x 1 = 1216 odd days--(B)

Total number of odd days = (A) + (B) = 768 + 1216 = 1984 odd days in 1600 years

and 1984 is not divisible by 7 !

Am I making a mistake ? If yes,what is it ?

Answer 1

Remember that a year divisible by $400$ is a leap year. Although $2100$ will not be a leap year. $2400$ will be.

So in $400$ years there are precisely $97$ leap years.

And yes, the calendar repeats every $400$ years, so the number of days in $1600$ years is divisible by $7$. For $400$ years, to the $(400)(364)$ days, just add $400+100-3$ (ordinary advance by $1$ day, plus 100 for the leap years sort of, minus $3$ for the multiples of $100$ that are not multiples of $400$). Then multiply by $4$. So for $1600$ we get $4(500-3)$ "additional" days. Your $1984$ was essentially computed correctly, except that we need $4$ additional days for the $4$ multiples of $400$.

Answer 2

While your definition of "odd days" is unclear to me, what I get is that if the number of days is a multiple of $7$, the number of odd days is $0$. In the following I'll therefore show that the number of days indeed is a multiple of $7$.

The rule for leap years is:

• If the year is divisible by $400$, it is a leap year.
• Otherwise, if it is divisible by $100$, it is not a leap year.
• Otherwise, if it is divisible by $4$, it is a leap year.
• Otherwise, it is not a leap year.

This means, the leap year rule has a $400$ year period. Therefore if we want to know how many days there are in $1600$ years, we can calculate the days in $400$ years, and multiply by $4$.

Now the number of days in $400$ years is $400\cdot 365 + \text{number of leap years}$. Now to calculate the number of leap years in $400$ years, we go through the above list in reverse order:

• Years divisible by $4$ are generally leap years, there are $400/4=100$ of them.
• However this way we have also counted the years divisible by $100$ (because they are all divisible by $4$). There are $400/100=4$ of them, which we therefore subtract, so we get $100-4=96$.
• However now we have also removed the years divisible by $400$, of which there's one in $400$ years. Therefore we have to add that one back, so we end up with $97$ leap years in $400$ years.

Therefore $400$ years have altogether $146\,097$ days. Now it is not hard to check that this number indeed is a multiple of $7$. Therefore there are no odd days in $400$ years, and thus also not in $1600$.