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Class act (Posted on 2002-05-02) Difficulty: 3 of 5
31 students in a row were numbered 1,2,...,31 in order. The teacher wrote down a number on the blackboard.

Student 1 said "the number is divisible by 1",
Student 2 said "the number is divisible by 2",
and so forth...until
Student 31 said "the number is divisible by 31".

The teacher remarked: "Very well pups, but two of you gave a wrong statement, and those two sit besides each other". Determine those two.

See The Solution Submitted by theBal    
Rating: 3.5000 (10 votes)

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Solution answer Comment 15 of 15 |

The two numbers can not be the factor of any other number in the set of positive integers between 1 and 31.

The following positive integers between 1 and 31, inclusively, have more than one number to which it is a factor:

  •  1 : 31 {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,
             16,17,18,19,20,21,22,23,24,25,26,27,
             28,29,30,31}
  •  2 : 15 {2,4,6,8,10,12,14,16,18,20,22,24,26,
             28,30}
  •  3 : 10 {3,6,9,12,15,18,21,24,27,30}
  •  4 :  8 {4,8,12,16,20,24,28}
  •  5 :  6 {5,10,15,20,25,30}
  •  6 :  5 {6,12,18,24,30}
  •  7 :  4 {7,14,21,28}
  •  8 :  3 {8,16,24}
  •  9 :  3 {9,18,27}
  • 10 :  3 {10,20,30)
  • 11 :  2 {11,22)
  • 12 :  2 {12,24)
  • 13 :  2 {13,26)
  • 14 :  2 {14,28}
  • 15 :  2 {15,30)
The following positive integers between 1 and 31, inclusively, can be divided into two of the above distinct divisors:
  • 18 : 3 and 6
  • 20 : 2 and 10 (or 4 and 5)
  • 21 : 3 and 7
  • 22 : 2 and 11
  • 24 : 2 and 12 (or 3 and 8; or 4 and 6)
  • 26 : 2 and 13
  • 27 : 3 and 9
  • 28 : 2 and 14 (or 4 and 7)
  • 30 : 2 and 15 (or 3 and 10; or 5 and 6)
This leaves the following numbers:
                 16, 17, 19, 23, 25, 29 and 31.
Only 16 and 17 are consecutive, thus the answer must by these two.

  Posted by Dej Mar on 2012-04-23 04:47:26
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