Joseph visits the club on every 5th day, Harsh visits on every 24th day, while Sumit visits on every 9th day. If all three of them met at the club on…

Correct Option (B)

To determine the next instance when all three individuals will meet at the club, it is essential to calculate the Least Common Multiple (LCM) of their respective visiting frequencies. The LCM represents the smallest number of days after which their individual visiting schedules will coincide.

• Joseph visits every 5th day.
• Harsh visits every 24th day.
• Sumit visits every 9th day.

The calculation for the LCM of $(5, 24, 9)$(5,24,9) is as follows:

• Prime factorization of $5 = 5$5=5
• Prime factorization of $24 = 2^3 \times 3$24=23×3
• Prime factorization of $9 = 3^2$9=32
• The LCM is derived by taking the highest power of all prime factors present: $LCM(5, 24, 9) = 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360$LCM(5,24,9)=23×32×5=8×9×5=360.

This indicates that all three individuals will meet again 360 days after their initial meeting on a Sunday.

To ascertain the specific day of the week, the concept of 'odd days' is applied. Odd days are the remainder obtained when the total number of days is divided by 7 (the number of days in a week).

• Number of odd days = $\frac{360}{7}$7360​
• $360 \div 7 = 51$360÷7=51 with a remainder of 3.
• Therefore, there are 3 odd days.

Counting 3 days forward from Sunday:

• Sunday + 1 day = Monday
• Sunday + 2 days = Tuesday
• Sunday + 3 days = Wednesday

Consequently, they will meet again on a Wednesday.

Incorrect Options:

Options A (Monday), C (Thursday), and D (Sunday) are incorrect. The precise calculation of the Least Common Multiple and the subsequent determination of odd days unequivocally establishes Wednesday as the next meeting day. Any deviation from this result would indicate an error in the calculation process.