Question

Prove that in every year, the 13 th day of some month occurs on a Friday?

Answer 1

See explanation...

Explanation:

Regardless of whether a year is a leap year or not, the months from March onwards have a fixed number of days each, so if we start counting with March 13th being day `0`, we have:

March 13th is day `0`
April 13th is day `31`
May 13th is day `61`
June 13th is day `92`
July 13th is day `122`
August 13th is day `153`
September 13th is day `184`
October 13th is day `214`

Modulo `7` these are:
`0, 3, 5, 1, 3, 6, 2, 4`

So March 13th, April 13th, May 13th, June 13th, August 13th, September 13th and October 13th will all be on different days of the week in any year (July 13th will be on the same day of the week as April 13th).

So one of them will be a Friday.

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Historical footnote

The year 1752 had a very strange calendar. 11 days (3rd - 13th) were dropped in September with the changeover from the Julian to the Gregorian calendar. As a result September had no 13th at all. Both March 13th and October 13th 1752 were Fridays, but there was no Tuesday 13th that year.