Another problem with ages

Question

Remember the famous three kids problem with the oldest one who plays the piano? Take a look at the next one:

A priest and a verger were travelling on a train carriage with three more passengers.
Then the priest said to the verger: "The product of those three passengers ages is 2450 and the sum is twice your age. How old are they?"
The verger made some mental calculation and replied to the priest that he didn't give him enough information to solve the problem.
"Very well!" - replied the priest - "One of all of you is older than everyone else!" - and the verger solved the problem.

How old are the priest, the verger and the three other passengers?

Answer 1

Solution

$$2450 = 2 \times 5^2 \times 7^2$$ which means that the three other passenger ages must be one of the following $$\{1,1,2450\}, \{1,2,1225\}, \{1,5,490\}, \{1,7,350\}, \{1,10,245\}, \{1,14,175\}, \{1,25,98\},$$ $$\{1,35,70\}, \{1,49,50\}, \{2,5,245\}, \{2,7,175\}, \{2,25,49\}, \{2,35,35\}, \{5,5,98\}, $$ $$\{5,7,70\}, \{5,10,49\}, \{5,14,35\}, \{7,7,50\}, \{7,10,35\}, \{7,14,25\}$$ which makes the sums of the ages one of the following $$2452, 1228, 496, 358, 256, 190, 124, 106, 100, 252, 184, 76, 72, 104, 82, \mathbf{64}, 54, \mathbf{64}, 52, 46$$ We notice that $64$ is the only possibility that appears twice so it is the only possibility that can confound the verger's calculation.

This means that the verger is 32

Furthermore, the passenger ages must either be $\{5,10,49\}$ or $\{7,7,50\}$.
Given the last statement the verger must know the priest's age and can only know the answer for definite if the priest is 49 (if the priest is older then the statement is false, younger and the verger cannot deduce the ages) which makes the passenger ages 7,7 and 50.