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A teacher wrote a large number on the board and asked the students to tell about the divisors of the number one by one.
The 1st student said, "The number is divisible by 2."
The 2nd student said, "The number is divisible by 3."
The 3rd student said, "The number is divisible by 4."
.
.
.
(and so on)
The 30th student said, "The number is divisible by 31.
The teacher then commented that exactly two students, who spoke consecutively, spoke wrongly.
Which two students spoke wrongly?
Answers
Since the two numbers are consecutive,
one of them is even and the other is odd.
Let’s say the two numbers were 2 and 3. If a number was not divisible by 2 and 3, then it would also not be divisible by every even number from 2 to 30 and every multiple of 3 from
3 to 30. Thus we have more than just two consecutive students who spokewrongwhichcontradicts what the teacher said. So, the two numbers cannot be 2 and 3
Using the same logic, we can conclude eliminate 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15.
Could it be, say 21 and 22? That will not work,because we know that it’s divisible by 7 and 3 and 2 an 11, so it must be divisible by 21 and 22.
Similarly, since we know the number is divisible by all the whole numbers from 2 through 15,
it must also be divisible by 18, 20, 21, 22, 24, 26, 28, and 30.
This leaves the following number pair: 16 and 17. Hence the two students who spoke wrong are the 15th and 16th students
The teacher wrote a very large number on the board .
Now the students are telling the divisors one by one .
The 1 st student tells 2 .
The 2 nd student tells 3 .
....
The 30 th student tells 31 .
The teacher comments "exactly two students, who spoke consecutively, spoke wrongly. "
Hence it is clear that the number of divisors of the number is less than or equal to 30 .
Clearly one of the number will be even and the other one will be odd .
Trial and Error .
The numbers would never be 20,21 .
This is because then the students who said 10,2 and 7,3 would have lied .
So cancel the numbers : 20 , 21 , 10 , 2 , 7 , 3 .
With the cancellation of 20 , we can cancel : 5, 4
Also the students cannot say 24 because then we will say that 8,3,6,4 are wrong .
Cancel more 24 , 8 , 6 , 3 are canceled .
Also 28 and 30 are of no use .
Cancel 15 , and also 31 .
Cancel 29 .
Cancel 25 and others like 14 , 11 , 22 .
Cancel 26 because of 13 and also cancel 18 and 19 .
Thus the number left with is 16 and 17 .
They are not divisible .
Hence the 15 th student and the 16 th student spoke wrong .