94 when divided by 7 gives remainder 3 hence today it must be Thursday, after 3 more we will get a Sunday, next Sunday will be after 3 + 7 = 10 days and so on so we will get Sunday after a day 7K + 3 days.

100 years contain 5 odd days.
? Last day of 1^st century is Friday.
200 years contain (5 x 2) = 3 odd days.
? Last day of 2^nd century is Wednesday.
300 years contain ( 5 x 3) = 15 = 1 odd day.
? Last day of 3^rd century is Monday
400 years contain 0 odd day.
? Last day of 4^rd century is Sunday.
This cycle is repeated that means last day of century is Friday, Wednesday, Monday, or Sunday and it repeats the cycle.

Here we have two cases :
Case (i) : If a month start with Thursday the month will have 5 Saturdays and 4 Sundays i.e total 9 weekends. In this case 31^st December will be Sunday hence 1^st January will be either Sunday (For non leap year) or Saturday ( For leap year) then January will have 9 or 10 weekends.
Case (ii) : If a month start with Sunday the month will have 4 Saturdays and 5 Sundays i.e total 9 weekends.
In this case 31^st December will be Wednesday hence 1^st January will be either Wednesday (For non leap year) or Tuesday ( For leap year) then January will have 8 or 9 weekends.
Hence number of weekends in January will be 8 or 9 or 10.

If there are 53 Saturday and Sunday that means year is a leap year and last day of year and last day of year is Sunday so 1^st day of January must be Saturday then January will have 10 weekends.

In a period of 100 years there are 23 or 24 leap years (as for century year it might be or might not be a leap leap year, as 1900 was not a leap year)
Number of odd days in a period of 100 years is 100 + 23 or 100 + 24 or 123/124 odd days.
Number of odd days in 100 years is when 123/124 divided by 7, remainder is 4 or 5. So next century should start with Monday or Tuesday.

In a period of 100 years there are 23 or 24 leap (as for century year it might be or might not be a leap leap year, as 1900 was not a leap year)
Number of odd days in a period of 100 years is 100 + 23 or 100 + 24 or 123/124 odd days.
Number of odd days in 100 years is when 123/124 divided by 7, remainder is 4 or 5. So next century should start with Friday or Saturday.

Last day of century can be Friday, Wednesday, Monday, or Sunday and it repeats the cycle. So 1^st day of next century must be either Saturday, Thursday, Tuesday, or Monday and so it must not be Wednesday, friday & Sunday

If previous year is leap year then calendar of May is similar to July

Each day of the week is repeated after 7 days. So, after 63 days, it will be Monday.

After 61 days, it will be Saturday.