### Instruction: What is the least positive integer value of $n$ such that $28n$ is divisible by $365$?### Response: To solve this problem, I need to find the smallest multiple of 28 that is also a multiple of 365.
I could try to list some multiples of 28 and see if any of them are multiples of 365, but that might take a long time.
A better way to approach this is to use the prime factorization of both numbers.
If I write 28 as $2^2 \cdot 7$ and 365 as $5 \cdot 73$, then I can see that for 28n to be divisible by 365, it must have at least one factor of 5 and one factor of 73 in its prime factorization.
This means that n must be a multiple of $5 \cdot 73 = 365$.
The smallest multiple of 365 is 365 itself, so the least positive integer value of n is 365.