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?
We can call the three integers in this problem $a,$ $b,$ and $c$. Then we have \begin{align*}
a &\equiv 7\pmod{12}, \\
b &\equiv 9\pmod{12}, \\
c &\equiv 10\pmod{12}.
\end{align*}Adding these congruences, we have \begin{align*}
a+b+c &\equiv 7+9+10 \\
&= 26\pmod{12}.
\end{align*}Therefore, $a+b+c$ has the same remainder as $26$ upon division by $12$. This remainder is $\boxed{2}$.
2
Level 1
Number Theory
Problem: 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?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the remainder when the sum of the three integers is divided by $12$, we first denote the three integers as $a$, $b$, and $c$. According to the problem, we have the following congruences:

1. $a \equiv 7 \pmod{12}$
2. $b \equiv 9 \pmod{12}$
3. $c \equiv 10 \pmod{12}$

Next, we can express each integer in terms of its remainder when divided by $12$:

- Let $a = 12k_1 + 7$ for some integer $k_1$.
- Let $b = 12k_2 + 9$ for some integer $k_2$.
- Let $c = 12k_3 + 10$ for some integer $k_3$.

Now, we calculate the sum of these integers:

$$ a + b + c = (12k_1 + 7) + (12k_2 + 9) + (12k_3 + 10) $$

Combining the terms, we have:

$$ a + b + c = 12(k_1 + k_2 + k_3) + (7 + 9 + 10) $$

Next, we compute the sum of the remainders:

$$ 7 + 9 + 10 = 26 $$

Thus, we can rewrite the sum as:

$$ a + b + c = 12(k_1 + k_2 + k_3) + 26 $$

To find the remainder when $a + b + c$ is divided by $12$, we focus on the term $26$. We need to compute $26 \mod 12$:

$$ 26 \div 12 = 2 \quad \text{(quotient)} $$
$$ 26 - 12 \times 2 = 26 - 24 = 2 \quad \text{(remainder)} $$

Therefore, the remainder when the sum of the three integers is divided by $12$ is $2$.<|END_OF_LATENT|> The final answer is $2$.