# Another problem with ages

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?

• Ah, sorry I forgot about the priest Jul 23, 2020 at 10:28
• @hexomino, bingo! ;)
– Pspl
Jul 23, 2020 at 10:29
• Very clever twist on a classic puzzle! Jul 23, 2020 at 10:30

$$2450 = 2 \times 5^2 \times 7^2$$ 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.
Furthermore, the passenger ages must either be $$\{5,10,49\}$$ or $$\{7,7,50\}$$.