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 a Sunday, then on which day will all three of them meet again?

Correct Answer: Wednesday

Step-by-step explanation:

Step 1: Find the least common multiple (LCM) of the visiting intervals.

Joseph visits the club every 5^th day.
Harsh visits the club every 24^th day.
Sumit visits the club every 9^th day.

To find out after how many days all three will meet together again, we need to calculate the LCM of 5, 24, and 9.

Let's write down the prime factorization of each number:

$5=5^1$

$24=2^3\times 3^1$

$9=3^2$

Taking the highest power of each prime factor involved (2, 3, and 5):

$\text{LCM}(5,24,9)=2^3\times 3^2\times 5^1=8\times 9\times 5=360$

So, all three of them will meet again after 360 days.

Step 2: Determine the day of the week after 360 days.

There are 7 days in a week. To find the day of the week 360 days after Sunday, we calculate the remainder when 360 is divided by 7:

$360÷7=51\text{weeks with a remainder of}3\text{days.}$

Since the remainder is 3, the three friends will meet on the 3^rd day after Sunday.

Counting 3 days forward from Sunday:

1 day after Sunday = Monday
2 days after Sunday = Tuesday
3 days after Sunday = Wednesday

Therefore, all three of them will meet again on a Wednesday.