Mr President "BONUS" day!

Number Theory Level 3

Because school days are now super tiring, all the students in "Holiday Country" are protesting to the Mr. President to make the weekend longer, or else they will not go to school anymore. Mr President then make a new rule so the students will be happy :D

The new rule made week become $8$ days and the name of the $8$ th day is " Bonus Day", but of course Mr. President cannot change the Earth's revolution, so a year is still $365$ days. But something happened... The biggest meteor in Galaxy, called "King Meteor" hit the sun (assume that it actually happened) and made leap years occur once in $6$ years.

But Mr President can only apply the new rule if the year modulo $3$ is $2$ (Example: if the year is $2015$ , then $2015 \equiv 2 \pmod{3}$ , so Mr. President apply the rule this year.

So if today is the first day of the year $N$ (you can call it New Year); $N \equiv 2 \pmod{3}$ ; today is Bonus Day ; this year is a leap year; and the first Bonus Day of the year $N + 140$ is the $B$ th day.

Find $B$ .

The answer is 1.

1 solution

Krisna Attayendra
Feb 3, 2015

To find the answer you just need to find the difference between the first day in each year.

if it is normal year then 365 mod 7 = 1 so you can say that the first day next year is tomorrow.

if it is BONUS Day year, so 365 mod 8 = 5 then you can say next year is day before yesterday.

If it is leap year so you just add 1 (Because there are 366 days in leap year).

I will not write the detail calculation, but if you did it wright you'll get (-2 * 47) + (1 * 94) + (1 * 24) = 24, and because year N+140 is also BONUS Day year then 24 mod 8 = 0, so the first year of year N+140 will also BONUS Day .

So the first BONUS Day is the first day and the answer is 1 .

Mr President "BONUS" day!

The answer is 1.

1 solution

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