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.

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