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A month always has 28 or more days, so as there are 7 days in a week there are always at least 4 Saturdays and four Sundays in a month. As most months have more than 28 days, there can be up to 5 Saturdays and up to 5 Sundays in a month.
There is no special name for that. Note that this can't happen for June - June only has 30 days. For months with 31 days, this is fairly common; on average, one in every seven years has (for example) a July with these properties.
3
In 2013, there are four.
When a month has 28 days, this can't happen. When it has 29 days (February of every leap year) it happens only when the first day of the month is a Saturday. When a month has 30 days, then if the first day of the month was a Saturday, you would also get 5 Sundays. To get 5 Saturdays and only 4 Sundays, then, means the first day of a 30-day month has to be a Friday. For a 31-day month, you need the first day of the month to be a Thursday, so that days 3, 10, 17, 24 and 31 are Saturdays and days 4, 11, 18 and 25 are Sundays. A rough calculation would suggest that the probability of getting the Saturday on the first day of a 29-day month is 1 in 7, that of getting it on the second day of a 30-day month is 1 in 7 and getting it on the third day of a 31-day month is 1 in 7. So any given month (except February on non-leap years) has a chance of about 1 in 7. 28-day months happen 97 out of every 4800 months in the current Gregorian calendar. So for the other 4703 months out of 400 years, there's a 1 in 7 chance. According to my calculations, it's a 14.00% chance of it happening in the long run, or about once every 7 months on average.