Saturday, October 02, 2004
What is it?
A teacher writes an integer less than 50,000 on the blackboard. One student states that the number is a multiple of 2; a second student states that the number is a multiple of 3; and so on and so forth until the twelfth student states that it is a multiple of 13. The teacher remarks that all except two of the students were right and moreover those two uttered wrongly spoke one after the other. What was the number that the teacher wrote on the blackboard?
Answer:
The teacher wrote 25,740 on the blackboard. Let N be the number that teacher wrote on the blackboard. Since all the students, except two who spoke one after the other, were correct; it can be deduced that N must be divisible by 1, 2, 3, 4, 5,6, 10, 11, 12 and 13. This is because if N is not divisible by 2, then it is not divisible by 4 also. If N is not divisible by 3, then it is not divisible by 6 also. If N is not divisible by 5, then it is not divisible by 10 also. And so on. All these leaves 7, 8 and 9 as the only possible numbers which do not divide N. Hence, there are 2 cases.
Case I: N is not divisible by 8 and 9. In this case, the smallest number divisible by all other numbers (i.e. Least Common Multiple) is 60,060. But the number is greater than the one written on the blackboard.
Case II: N is not divisible by 7 and 8. In this case, the smallest number divisible by all other numbers (i.e. Least Common Multiple) is 25,740. Thus, the teacher wrote 25,740 on the blackboard.
Answer:
The teacher wrote 25,740 on the blackboard. Let N be the number that teacher wrote on the blackboard. Since all the students, except two who spoke one after the other, were correct; it can be deduced that N must be divisible by 1, 2, 3, 4, 5,6, 10, 11, 12 and 13. This is because if N is not divisible by 2, then it is not divisible by 4 also. If N is not divisible by 3, then it is not divisible by 6 also. If N is not divisible by 5, then it is not divisible by 10 also. And so on. All these leaves 7, 8 and 9 as the only possible numbers which do not divide N. Hence, there are 2 cases.
Case I: N is not divisible by 8 and 9. In this case, the smallest number divisible by all other numbers (i.e. Least Common Multiple) is 60,060. But the number is greater than the one written on the blackboard.
Case II: N is not divisible by 7 and 8. In this case, the smallest number divisible by all other numbers (i.e. Least Common Multiple) is 25,740. Thus, the teacher wrote 25,740 on the blackboard.
# posted by padmaram @ 8:10 PM 
