Pr. 13. In a class, 25 students were lined up. The teacher wrote a number on the board.
• The first student said the number was divisible by 1.
• Student number 2 said it was divisible by 2.
• Student 3 said it was divisible by 3.
...
• This went on until the 25th student.
The teacher said everyone was correct except two boys, named Jack and Jill, who were wrong. They were standing one immediately after the other in the line.
What are their positions in the line?
I had no idea how to start, so I listed the factors and multiples under 25 of the numbers under 25:
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
2: 1,4,6,8,10,12,14,16,18,20,22,24
3: 1,6,9,12,15,18,21,24
4: 1,2,8,12,16,20,24
5: 1,10,15,20,25
6: 1,2,3,12,18,24
7: 1,14,21
8: 1,2,4,16,24,
9: 1,3,18
10: 1,2,5,20
11: 1,22
12: 1,2,3,4,6,24
13: 1
14: 1,2,7
15: 1,3,5
16: 1,2,4,8
17: 1
18: 1,2,9
19: 1
20: 1,2,4,5
21: 1,3,7
22: 1,2,11
23: 1
24: 1,2,3,4,6,8,12
25: 1,5
I thought I could choose 17 and 19, but they aren't consecutive!
How shall I attempt and solve this question?
If all of the responses had been correct, then the number would have to be a multiple of
$$2^4 \times 3^2 \times 5^2 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23.$$
The obvious try, which works, is that neither $(16)$ nor $(17)$ divides the number.