I want to find the sum of the three integers modulo $24$, which means I only care about their remainders when divided by $24$. I can add up the remainders of the three integers: $10 + 4 + 12 = 26$. But this sum is larger than $24$, so I need to reduce it modulo $24$. That means I need to find the remainder of $26$ when divided by $24$. I can do this by subtracting $24$ from $26$: $26 - 24 = 2$. Therefore, the remainder of the sum of the three integers when divided by $24$ is $2$.


When three positive integers are divided by $24$, the remainders are $10,$ $4,$ and $12,$ respectively.

When the sum of the three integers is divided by $24$, what is the remainder?