### Instruction: What is the smallest positive integer $n$ such that $3n \equiv 1356 \pmod{22}?$.### Response: To find $n$, I need to solve the linear congruence $3n \equiv 1356 \pmod{22}$.
One way to do this is to use the Euclidean algorithm to find the inverse of $3$ modulo $22$.
Then I can multiply both sides of the congruence by the inverse to get $n \equiv 1356 \cdot 3^{-1} \pmod{22}$.
To find the inverse of $3$ modulo $22$, I need to find integers $x$ and $y$ such that $3x + 22y = 1$.
This is equivalent to finding the greatest common divisor of $3$ and $22$ and expressing it as a linear combination of $3$ and $22$.
I can use the Euclidean algorithm to do this.
First, I divide $22$ by $3$ and get a quotient of $7$ and a remainder of $1$.
This means that $22 = 3 \cdot 7 + 1$, or equivalently, $1 = 22 - 3 \cdot 7$.
So I can take $x = -7$ and $y = 1$ as one solution.
This also means that $-7$ is the inverse of $3$ modulo $22$.
To find a positive inverse, I can add $22$ to $-7$ and get $15$.
Therefore, $3^{-1} \equiv 15 \pmod{22}$.
Now I can multiply both sides of the original congruence by $15$ and get $n \equiv 1356 \cdot 15 \pmod{22}$.
To simplify this, I can divide $1356$ by $22$ and get a quotient of $61$ and a remainder of $14$.
This means that $1356 \equiv 14 \pmod{22}$.
So I can replace $1356$ with $14$ and get $n \equiv 14 \cdot 15 \pmod{22}$.
Multiplying $14$ by $15$, I get $210$.
Dividing $210$ by $22$, I get a quotient of $9$ and a remainder of $12$.
This means that $210 \equiv 12 \pmod{22}$.
So I can replace $210$ with $12$ and get $n \equiv 12 \pmod{22}$.
This is the simplest form of $n$ modulo $22$, and it is also the smallest positive integer that satisfies the congruence.