TIFR CSE 2020 | Part A | Question: 10

Question

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)?

• A. Sunday
• B. Monday
• C. Wednesday
• D. Friday
• E. None of the others

Answer 1

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

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

Answer 2

yes Sunday and the option B is correct. For exactly FOUR Monday and Friday and the complete week 31/7=4 weeks and we have to find out otehr consecutive days (which doesnot include Monday and Friday)beyond complete FOUR weeks. Hence the will be TUESDAY,WEDNSEDAY ,THURSDAY on 29 th ,30 th,31st of January of that year.

Hence 28 th will be Monday,and so on 21 st will be Monday.Hence 20 th will be SUNDAY and the option A is correct.