lf the 1st day of a month falls on a Monday, on which day will the 30th fall?
Answers
Answer:
If the month is of 30 days then Tuesday
or if it is of 31 days then wednesday
Step-by-step explanation:
PLEASE MARK BRAINLIEST
Answer:
Well if the 30th of the month falls on a given day, let's normalize for Monday and January, then on most years February doesn't change anything since it's 28 days and we're working modulo 7. (Please ask to clarify in case it isn't familiar)
Step-by-step explanation:
Then, look we have one more day in January and then 30 days in March, that's 4 mod 7 or Thursday. Afterwards things look like this :
January = 1 mod 7
March = 4 mod 7 (5)
April: 1 + 30 + 4 = 0 mod 7 (1)
May: 0 + 30 + 0 = 2 mod 7 (3)
June: 1 + 30 + 2 = 5 mod 7 (6)
July: 30 + 5 = 0 mod 7 (1)
August: 1 +30 + 0 = 5 mod 7 (6)
September : 1 + 30 + 5= 1 mod 7 (2)
October : 30 + 1 = 3 mod 7 (4)
November : 1 + 30 + 3 = 6 mod 7 (0)
December : 30 + 6 = 1 mod 7 (2)
Now, counting we can see that the day with the most frequency is the day of the first 30th. If you have a leap year shift everything by a day every month starting with March. As you can see from the parentheses, you still get the same results.
Finally, in a four year cycle(where the fourth year is a leap year) you get the following results assuming year 1 we had starting date and most frequent date equal to 1 mod 7.
Year 1 : 1 mod 7
Year 2: 4 mod 7
Year 3: 0 mod 7
Year 4: 4 mod 7
And then, let's look at what happens over a longer period of time
Year 1 : 1 mod 7
Year 2: 4 mod 7
Year 3: 0 mod 7
Year 4: 4 mod 7
Year 5 : 0 mod 7
Year 6: 3 mod 7
Year 7: 6 mod 7
Year 8: 3 mod 7
Year 9: 6 mod 7
Year 10: 2 mod 7
Year 11: 5 mod 7
Year 12: 2 mod 7
Notice any patterns? This is what makes this question beautiful, over a four year period there is actually a day which is more likely or 3 days after the first 30th day in that cycle. However, every time we consider a new cycle of 4 years we start off a day earlier in the calendar than in the previous cycle, and therefore our most likely day is also a day earlier than in the previous cycle. So finally, the answer is that for long enough periods of time you have equal likelihood of any date of the month falling on a given day of the week