Members of the lake shawnee club pay $40 per summer season pass plus $7.50 each time they rent a boat. nonmembers pay $12.50 each time they rent a boat. how many times would both a member and a nonmember have to rent a boat in order to pay the same amount?

Members: $40 for season pass + $7.50 per rent
Non-members: $12.50 per rent

The simplest way to do it is to find some of the first iterations (multiples) of 12.50:

12.5 * 1 = 12.5
12.5 * 2 = 25
12.5 * 3 = 37.5
12.5 * 4 = 50
12.5 * 5 = 62.5
12.5 * 6 = 75
12.5 * 7 = 87.5
12.5 * 8 = 100

Now we do the same with each 7.50 and add 40 to each result:

7.50 * 1 + 40 = 47.50
7.50 * 2 + 40 = 15 + 40 = 55
7.50 * 3 + 40 = 22.5 + 40 = 62.5
7.50 * 4 + 40 = 30 + 40 = 70
7.50 * 5 + 40 = 37.5 + 40 = 77.5
7.50 * 6 + 40 = 45 + 40 = 85
7.50 * 7 + 40 = 52.5 + 40 = 92.5
7.50 * 8 + 40 = 60 + 40 = 100

As we can see, both members and non-members can pay $65.50 after renting a boat specific number of times. For non-members it's after 5 rents (as 62.5 is 5th iteration of 12.5) and for members it's after 3 rents.

It's also interesting that they'll pay the same amount ($100) after 8 rents, no matter if they bought the season pass or not.

31 is prime. and it’s prime factorization is just 31.