I want to find the sum of the three integers modulo $12$. That means I only care about their remainders when divided by $12$. So I can ignore the quotient parts and just add up the remainders. The sum of the remainders is $7 + 9 + 10 = 26$. Now I need to find the remainder when $26$ is divided by $12$. I can use long division or repeated subtraction to do that. I see that $26 = 12 \times 2 + 2$. So the remainder is $2$. 
When three positive integers are divided by $12$, the remainders are $7,$ $9,$ and $10,$ respectively.

When the sum of the three integers is divided by $12$, what is the remainder?