Prove that in each year, the 13th day of some month occurs on a Friday Prove that in each year, the 13th day of some month occurs on a  Friday.
No clue... please help!

In fact, every year will contain a Friday the 13-th between March and October (so leap years don't enter into it).
If March 13 is assigned $0 \pmod 7$, then the other moduli occur as indicated below:
$$(\underbrace{\underbrace{\underbrace{\underbrace{\underbrace{\underbrace{\overbrace{31}^{\text{March}}}_{3 \pmod 7},\overbrace{30}^{\text{April}}}_{5 \pmod 7},\overbrace{31}^{\text{May}}}_{1 \pmod 7},\overbrace{30}^{\text{June}},\overbrace{31}^{\text{July}}}_{6 \pmod 7},\overbrace{31}^{\text{August}}}_{2 \pmod 7},\overbrace{30}^{\text{September}}}_{4 \pmod 7})$$
