Since each week consists of 7 days, we will be working in modulo 7.  Let the first day of the year, January 1, be represented by 1.  Then the seventh day of the month will be represented by 0 in mod 7, since
Each time the cycle repeats, every weekday is represented by the same number.  For example, if the first day of the year falls on a Thursday, then every Thursday throughout that year will be represented by 1.
Therefore, in a calendar year which begins on a Thursday, every Thursday will be represented by a 1.  Similarly, all other days will be also be represented by the same number.

We would like to show that for any given year there must be at least one month and at the most three months for which the 13th of the month falls on a Friday.

Friday the 13th

Given two distinct integers a and b, a is said congruent to b in mod n if the difference of a and b is some multiple of n.
Let n = 10 and a = 17.
Then b must be 7, since
17 - b = 10
b = 7
Similarly,

Show that for any year there must be at least one month and at most three months for which the 13th of the month falls on Friday.
First, let us examine a non-leap year situation.  Again, let the first day of the year be represented by 1.  It is irrelevant which weekday this happens to be.  Label each day of every month as determined by the cycle described above; in other words, consider each day of every month in modulo 7.  Keep the days separated by the months in which they fall.
The Problem
On this calendar, the 13th day of each month has been highlighted.  Therefore, we have discovered the following congruences:

A complete cycle consists of the numbers 0 - 6, where each number is representing a specific day of the week.  As we can see by the calendar, each day of the week falls on the thirteenth at least once during the calendar year. Some weekdays only occur once; when 13 is congruent to 0, 1, or 3.  Others occur twice; when 13 is congruent to 4, 5, or 6. One day of the week occurs three times throughout the year; when 13 is congruent to 4.  Since each weekday in the cycle is represented, we know that Friday the 13th must occur at least one time and at most three times throughout the year.
Richard Francisco & Molly McKee