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Will Friday the 13th occur every year?

i decided i need a property for Fri the 13th to make my proof easier. i chose the fact that if Friday is the 13th, then Sunday is the 1st. From there, i set out to prove that every year, at least one Sunday will be the first of some month, thus proving the existence of Fri the 13th.

CLAIM: It doesn't matter what day the year starts on because no matter what day the year begins on, each day of the week is the first of some month (which would complete my proof). for this proof, i used modular arithmetic. its a lot more simple than it looks..

Let 0=Su, 1=M, 2=Tu, 3=W, 4=Th, 5=F, 6=Sa (so, I'm working in mod 7)

CASE 1: Non-Leap Year

Jan 1=0mod7

Feb 1=31=3mod7

Mar 1=31=3mod7

Apr 1=34=6mod7

May 1=36=1mod7

Jun 1=32=4mod7

Jul 1=34=6mod7

Aug 1=37=2mod7

Sep 1=33=5mod7

Oct 1=63=0mod7

Nov 1=31=3mod7

Dec 1=33=5mod7

This is the initial rotation of 1st's. Note that all of the numbers 0-6 (mod 7) are represented. This means that each day starts some month. CASE 2: Leap Year can also be studied based on CASE 1. Note that between Feb-Dec, which is what a Leap Year alters by one day, all of the numbers 0-6 are represented. Therefor, in a given year, Leap or Not, each day of the week falls on the first of some month, which implies that some month will always have a Friday the 13th. ■

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Q: Does every year contain at least one Friday the 13th?

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