Bài Toán chia hết trong đề thi APMOPS 2013


Question 1: A 5-digit number written in the form \overline{24abc} has the last three digits unknown. If this number is divisible by 3,4 and 5 respectively, find the greatest possible value that \overline{abc} can take.

Answer: 960.

Solution:

This number (A) is divisible by 4 and 5. Thus c must be 0. We can start with the case a=9. Because A is divisible by 4, we imply that there are only four possible cases for b: 2,4,6,8. Now it is not difficult to conclude that the number 960 is our answer (notice that A is divisible by 3).

Question 2: The sum of 10 positive integers, not necessary distinct, is 1001. If d is the greatest common divisor of the 10 numbers, find the maximum possible value of d.

Solution:

Assume that these 10 numbers are a_1,a_2,...,a_{10} and a_i=dk_i \forall i=1,...,10. We have d(k_1+k_2+...+k_{10})=1001=7.11.13.

Notice that k_1+...+k_{10} \geq 10 and it’s smallest possible value is 11 (it is possible because we can take k_1=k_2=...=k_9=1 and k_{10}=2). In this case we have d=7.13=91. Hence, the maximum possible value of d is 91.



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