At a large picnic, there were 45 dishes served altogether. Every 3 people shared a dish of Cole slaw between them. Every 4 people shared a dish of potato salad between them. Every 6 people shared a dish of hot dogs between them. How many people were at the picnic?

If every 3 people shared a dish of Cole slaw between them, every 4 people shared a dish of potato salad between them and every 6 people shared a dish of hot dogs between them, then you can find the minimal number of people that can do such thing. Mathematically this is the least common multiplier LCM.

Since

3=3;
4=2·2;
6=2·3,

then LCM(3,4,6)=2·2·3=12.

Suppose at the picnic were 12 people. If every 3 people shared a dish of Cole slaw between them, then 12 : 3 = 4 such dishes were served. If every 4 people shared a dish of potato salad between them, then 12 : 4 = 3 such dishes were served. If every 6 people shared a dish of hot dogs between them, then 12 : 6 = 2 such dishes were served. In total, 4 + 3 + 2 =9 dishes.

You know there were 45 dishes served altogether. 45 is 5 times more than 9, then the number of people at the picnic is 5 times more than 12. Therefore, there were 12·5=60 people at the picnic.