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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.

Michael Hardy
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Carli
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  • What have you tried? You are more likely to get an answer on StackExchange if show the work that you have already done (even if it isn't likely to lead to a solution) In this case, one approach is to consider the factors of the three numbers on the whiteboard.If all of the HCF's are correct, then any factor of any of the HCF's must be a common factor of two of the numbers. – Dylan Jun 26 '15 at 08:48
  • Have you any thoughts of your own on this problem? What have you tried? – Mark Bennet Jun 26 '15 at 08:50
  • I've edited my post to include my current working. Thank you for this tip :) – Carli Jun 26 '15 at 09:02
  • Suppose the numbers were $18, 27, 12$ - in pairs the HCFs are $9, 6, 3$ and only one of the HCFs is divisible by $9$ – Mark Bennet Jun 26 '15 at 09:08

1 Answers1

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HINT: The first and third highest common factors are divisible by $4$; the second one isn't.

Brian M. Scott
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