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In a certain year, there were exactly four Fridays and exactly four Mondays in January. On what day of the week did the $20^{th}$ of the January fall that year (recall that January has $31$ days)?

1. Sunday
2. Monday
3. Wednesday
4. Friday
5. None of the others

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+1 vote

January has 31 days, no. of complete weeks in Jan $=\left \lfloor 31/7 \right \rfloor =4$

Then remaining days $31-7(4)=3$. Since mentioned there are exactly 4 Mondays and 4 Fridays then these Mondays and Fridays are already covered in the 4 complete weeks.Hence for these $3$ days we need 3 consecutive days other than Monday and Friday.

The only 3 consecutive days other than Monday and Friday is :

$29^{th} Jan \rightarrow Tuesday$

$30^{th} Jan \rightarrow Wednesday$

$31^{st} Jan \rightarrow Thursday$

then the $4$ Mondays are :

$28^{th} Jan \rightarrow Monday$

$21^{st} Jan \rightarrow Monday$

$14^{th} Jan \rightarrow Monday$

$7^{th} Jan \rightarrow Monday$

Thus

$20^{th} Jan \rightarrow \textbf{Sunday}$

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