Alicia has $n$ candies, where $n$ is a positive integer with three digits. If she buys $5$ more, she will have a multiple of $8$. If she loses $8$, she will have a multiple of $5$. What is the smallest possible value of $n$?

123 + 5 = 128 multiple of 8

123 - 8 = 115 multiple of 5

8 - 5 = 3

LCM (5, 8) = 40

(n + 5) mod 8 = 0

(n - 8) mod 5 = 0 , solve for n

n = 40D + 3, for D=0, 1, 2, 3......etc. Since n must the smallest positive 3-digit integer, then we have:

n = 40*3 + 3

n = 120 + 3

n = 123