### Instruction: 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?### Response: 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 $2$.