Question 48
Vukasin, Dimitrije, Dusan, Stefan and Filip asked their professor to guess three consecutive positive integer numbers after they had told him these (true) sentences:
Vukasin: "Sum of the digits of one of them is a prime number. Sum of the digits of some of the other two is an even perfect number ($n$ is perfect if $\sigma(n) = 2n$). Sum of the digits of the remaining number is equal to the number of its positive divisors."
Dimitrije: "Each of these three numbers has no more than two digits 1 in its decimal representation."
Dusan: "If we add 11 to one of them, we obtain a square of an integer."
Stefan: "Each of them has exactly one prime divisor less than 10."
Filip: "The 3 numbers are square-free."
Their professor gave the correct answer. Which numbers did he say?
Vukasin: "Sum of the digits of one of them is a prime number. Sum of the digits of some of the other two is an even perfect number ($n$ is perfect if $\sigma(n) = 2n$). Sum of the digits of the remaining number is equal to the number of its positive divisors."
Dimitrije: "Each of these three numbers has no more than two digits 1 in its decimal representation."
Dusan: "If we add 11 to one of them, we obtain a square of an integer."
Stefan: "Each of them has exactly one prime divisor less than 10."
Filip: "The 3 numbers are square-free."
Their professor gave the correct answer. Which numbers did he say?