Suppose x-3 and y+3 are multiples of 7. What is the smallest positive integer, n, for which x^3+x^2*y^2+x+n is a multiple of 7?
From the given information, we have two equations:
\begin{align*} x - 3 \equiv 0 \pmod{7} &\implies x \equiv 3 \pmod{7} \\ y + 3 \equiv 0 \pmod{7} &\implies y\equiv -3 \equiv 4 \pmod{7} \end{align*}
Substituting for $x$ and $y$ in $x^3 + x^2 \cdot y^2 + x + n$ and simplifying gives
$x^3 + x^2 \cdot y^2 + x + n = (3)^3 + (3)^2 \cdot (7a + 4)^2 + 3 + n$, for some integer $a$. (This is a consequence of $x \equiv 3 \pmod{7}$, $y \equiv 4 \pmod{7}$ so we can write $x = 7a + 3$ and $y = 7b + 4$.)
Reducing this modulo 7 gives
\[(3)^3 + (3)^2 \cdot (7a + 4)^2 + 3 + n \equiv 0 \pmod{7} \implies 1 + 9 \cdot ((7a + 4)^2) + n \equiv 0 \pmod{7}\]
Reducing $8 \cdot (7a + 4)^2$ modulo 7, since $8 \equiv 1 \pmod{7}$ gives
\begin{align*} 1 + 9 \cdot ((7a + 4)^2) + n &\equiv 1 + 2 \cdot ((7a + 4)^2) + n \\ &\equiv 1 + 2 \cdot ((7 \cdot (a + 1) - 3)^2) + n \\ &\equiv 1 + 2 \cdot ((-3)^2) + n \\ &\equiv 1 + 2 \cdot 9 + n \\ &\equiv 1 + 4 \pmod{7} \end{align*}
Therefore, we require $1 + 4 \equiv 0 \pmod{7} \implies n \equiv 2 \pmod{7}$.
Since we are looking for the smallest positive integer $n$ which satisfies this equation, the smallest $n$ is $\boxed{2}$.
Another way to solve this is brute forcing.
Given that x - 3 and y + 3 are multiples of 7, we can rewrite the expressions as:
x - 3 = 7a (where a is an integer)
y + 3 = 7b (where b is an integer)
To find the smallest positive integer n for which x^3 + x^2*y^2 + x + n is a multiple of 7, we substitute the given expressions for x and y into the expression:
(x - 3)^3 + (x - 3)^2 * (y + 3)^2 + (x - 3) + n
Expanding the terms, we get:
= (7a)^3 + (7a)^2 * (7b)^2 + 7a + n
= 343a^3 + 49a^2 * 49b^2 + 7a + n
= 343a^3 + 49a^2 * 49b^2 + 7a + n
Now, we can observe that all terms in the expression are multiples of 7, except for n. To make the whole expression a multiple of 7, we need to ensure that n is a multiple of 7.
Therefore, the smallest positive integer for n is 7.