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(Got this from my facebook news feed.)

A friend of mine, upset after the earlier housing number conundrum, decided to make me guess his. He said:

  1. If my house number is a multiple of 3, it's from 50 to 59.
  2. If it is not a multiple of 4, then it is a number from 60 to 69.
  3. If it is not a multiple of 6, then it is a number from 70 to 79.

What's the house number?

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4 Answers 4

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The house number must be 76.

Either the number is a multiple of 3 or it is not. If it is then it is between 50 and 59, if it is not, then it is not a multiple of 6 either, and thus it is between 70 and 79.

Therefore we know it is not between 60 and 69, and therefore it is a multiple of 4.

If it is between 50 and 59, and is a multiple of 4 it must be 52 or 56. But neither of these are multiples of 3. Therefore the number is not between 50 and 59, but between 70 and 79.

Of the numbers between 70 and 79, it can be 72 or 76. Since 72 is a multiple of 6 however, this leaves 76 as the only possible solution.

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    $\begingroup$ Could you show your work? $\endgroup$
    – generalcrispy
    Commented Nov 7, 2014 at 14:43
  • $\begingroup$ @generalcrispy, working has been added. $\endgroup$
    – Kenshin
    Commented Nov 7, 2014 at 14:50
  • $\begingroup$ Gotta give it to you, you edited first. $\endgroup$
    – generalcrispy
    Commented Nov 7, 2014 at 16:46
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Writing the long answer:

If it's a multiple of 3 is between 50 and 59: 51, 54, 57, but then it should be also a multiple of 4 and 6 and none of them are.
If it's not a multpile of 4 it's between 60 and 69: 60, 61, 62, 63, 64, 65, 66, 67, 68, 69; 63, 66, and 69 are multiple of 3 and not vaild, while 61, 62, 65, 67 and 69 are not multiple of 6 and thus not valid either. The number must be between 70 and 79.
72, 75 and 78 are multiple of 3 and not valid. 70, 71, 73, 74, 77 and 79 are not multiple of 4 and not valid. The only remaining number is 76.

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The house number can't be a multiple of 3: It would be from 50 to 59, so it would be one of 51, 54 or 57, all these are not multiples of four, so the house number would also be from 60 to 69 which isn't possible.

Since it's no multiple of 3, it's no multiple of 6 either, so its from 70 to 79. Of these only 72 and 76 are multiples of four, but 72 is a multiple of 3 which the house number isn't, that leaves only 76.

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The answer is

76

If the answer were between 50 and 59 then it must be a multiple of 3 (by rule 1), but also a multiple of 4 and 6 (to negate the other 2 rules). There are no such numbers in this range.

If the answer were between 60 and 69, then it must not be a multiple of 3, but must be a multiple of 6. There are no numbers that can be a multiple 6 and not a multiple of 6. So that rules out this range.

To find the number between 70 and 79, one must find a number that is not a multiple of 6, nor a multiple of 3, but is a multiple of 4.

Finally:

76 is the product of 2 x 2 x 19 - thus it is a multiple of 4, but not 3 or 6 (as needed).

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