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Class act (Posted on 2002-05-02 )

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.

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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