What are the next numbers in the following sequences?
a: 77, 143, 221, 323, 437
b: 1, 5, 32, 288, 3413
c: 827, 23, 11, 6, 3
d: 2, 3, 5, 11, 31
e: 100, 121, 144, 202, 244

Suppose you have five bags filled with the same type of coins and a weigthing machine. One of this bags is completely filled with fake coins, the other bags contain only real ones. A real coin weights 10 gram and a fake one 11 gram. What is the minimal number of times you need to use the weighting machine in order to find the bag with the fake coins?

A man would like to take a new health insurance. An officer taking care of these matters says to the man: "Please tell me how many children you have." The man answers: "I have three of them." The officer: "What are the ages of your children?". The man answers: "The product of the ages is equal to 36." The officer replies: "This is not enough information Sir!". "Sorry that I was a little bit unclear, but the sum of the ages is equal to the number of shops in front of your office," says the man. The officer: "This still isn't enough information Sir!". The man replies: "My oldest child loves chocolate." The officer: "Thanks for your cooperation, I now know the ages."
Are you as smart as the officer? Then give the ages of the children.