100 each

Number Theory Level 5

A Rich Man once decided to hold a party at his place. He invited 100 guests and arranged for 100 plates of food. The guests were of three social groups: A, B and C.
The members of group A said that they will eat 2 plates each,
The members of group B said that they will eat 4 plates each.
Seeing that the rich man was getting worried, the members of Group C volunteered to eat 4 persons to a plate, thereby solving the man's problem.

Now, no plates were left and all the guests had eaten food.
How many members of each group were there?

Enter the answer as \(0\) if you think no answer is possible.
Enter the answer as \(a*b*c\) if there is a unique answer.
Enter the answer as the difference between the highest and lowest values of \(a*b*c\) if there are more than \(1\) answers.

Details and assumptions:

  • There is at least 1 member of each group.

  • Each plate in case of group C must be used by exactly 4 members i.e. you cannot have only 2 members eating from the last plate.

  • \(a,b,c\) refer to number of members of A,B and C present at the party.


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