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To find the smallest four-digit number divisible by both 6 and 15, we first determine the least common multiple (LCM) of 6 and 15.
The LCM is obtained by taking the highest power of each prime appearing in the factorizations:
Now, we seek the smallest four-digit number divisible by 30. The smallest four-digit number is 1000.
The smallest integer greater than or equal to 33.333 is 34. Thus, the smallest multiple of 30 that is a four-digit number is:
Therefore, the smallest four-digit number divisible by both 6 and 15 is:
The sum of all positive divisors of $ 720 $ is given by the product:
Thus, the sum of all even positive divisors of $ 720 $ is:
Let $ a = 5m $, $ b = 5n $, where $ m, n $ are positive integers. Then:
Try $ d = 10 $: then $ x + y = 2 $, so $ x = y = 1 $, which are coprime. Valid.