Calvin is organizing a dinner party for a group of friends. He has prepared 45 chicken dumplings and 75 shrimp wantons. He manages to split the chicken dumplings and shrimp wantons evenly amongst everyone, so that everyone got the same number of chicken dumplings as each other, and similarly for the shrimp wantons. What is the most number of people that could be at the party?

Let the number of people who attended the dinner party be N. Then, from the conditions of the question, 45 is a multiple of N and 75 is a multiple of N. Hence, N is a factor of both 45 and 75, so the largest possible value of N would be the greatest common divisor of 45 and 75.

We can calculate that gcd(45,75)=15.

15

To find the maximum number of people that could be at the party, we need to determine the common factor of both the number of chicken dumplings and shrimp wantons that can evenly divide them.

The factors of 45 are: 1, 3, 5, 9, 15, 45.
The factors of 75 are: 1, 3, 5, 15, 25, 75.

By comparing the two lists of factors, we can see that the common factors are 1, 3, 5, and 15.

Since everyone at the party must receive an equal number of chicken dumplings and shrimp wantons, the maximum number of people would be the highest common factor of 45 and 75, which is 15.

Therefore, the most number of people that could be at the party is 15.