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When it is necessary to allocate a week to a single month, the rule for first week of the year might be applied, although ISO 8601-1 does not consider this case explicitly. The resulting pattern would be irregular. There would be 4 months of 5 weeks per normal, 52-week year, or 5 such months in a long, 53-week year.
Similarly, the last ISO week of a year may have up to three days that are actually in the Gregorian calendar year that is starting; if three, they are Friday, Saturday, and Sunday. The Thursday of each ISO week is always in the Gregorian calendar year denoted by the ISO week-numbering year. Examples: Monday 29 December 2008 is written "2009-W01-1"
For Saturday, this ends up being equivalent to the week-date rule from ISO 8601 which ensures that the first week of the year contains four or more days (i.e. its majority) of that year, which includes the first Thursday and January 4. In this scenario, fiscal years would end on the following days: 2023 September 2; 2024 August 31; 2025 August 30
The calendar year has 13 months with 28 days each, divided into exactly 4 weeks (13 × 28 = 364). An extra day added as a holiday at the end of the year (after December 28, i.e. equal to December 31 Gregorian), sometimes called "Year Day", does not belong to any week and brings the total to 365 days.
Weeks are generally referred to by the date of some day within that week (e.g., "the week of May 25"), rather than by a week number. Many holidays and observances are identified relative to the day of the week on which they are fixed, either from the beginning of the month (first, second, etc.) or end (last, and far more rarely penultimate and ...
The last option is, oviously, the use case where you would apply the ISO week rule to months. This does not need to be visible as clearly, imagine continuously scrolling week rows with a month (and year) label which is not large enough to bear any two names at once. Options one and two are prevalent in paper calendars focused on months.
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ISO 8601 includes a specification for a 52/53-week year. Any year that has 53 Thursdays has 53 weeks; this extra week may be regarded as intercalary. The xiuhpōhualli (year count) system of the Aztec calendar had five intercalary days after the eighteenth and final month, the nēmontēmi, in which the people fasted and reflected on the past year.