Bài Toán chia hết trong đề thi APMOPS 2013Posted: September 21, 2013
Question 1: A 5-digit number written in the form has the last three digits unknown. If this number is divisible by 3,4 and 5 respectively, find the greatest possible value that can take.
This number () is divisible by 4 and 5. Thus must be 0. We can start with the case . Because is divisible by 4, we imply that there are only four possible cases for : 2,4,6,8. Now it is not difficult to conclude that the number 960 is our answer (notice that is divisible by 3).
Question 2: The sum of 10 positive integers, not necessary distinct, is 1001. If is the greatest common divisor of the 10 numbers, find the maximum possible value of .
Assume that these 10 numbers are and We have
Notice that and it’s smallest possible value is 11 (it is possible because we can take and ). In this case we have Hence, the maximum possible value of is 91.