Three positive integers are written on a whiteboard.
- David calculated the HCF of two of them and obtained 1 000 004
- Rose calculated the HCF of two of them and obtained 1 000 006
- Stephen calculated the HCF of to of them and obtained 1 000 008
Emily is sure that at least one of her friends made a mistake despite the fact that they calculated the HCF of different numbers. Is she right?
[ A concise proof would be greatly appreciated ]
EDIT: I have tried proving that Stephen has made a mistake due to the fact that his HCF is divisible by 9 but neither Rose nor David's HCFs are even though at least one of them must share a common integer that is a product of 9. I'm not sure if this is heading in the right direction though.
HINT: The first and third highest common factors are divisible by $4$; the second one isn't.