+39 Suppose x-3 and y+3 are multiples of 7. What is the smallest positive integer, n, for which x^2+xy+y^2+n is a multiple of 7? By the way it's not 4
We have that x−3=7a and y+3=7b for some integers a and b. Then \begin{align*} x^2+xy+y^2+n &= (x-3)^2 + 2(x-3)(y+3) + (y+3)^2 + n \ &= 49a^2 + 2(7a)(7b) + 49b^2 + n \ &= 49(a^2+7ab+b^2) + n \ &\equiv n \pmod{7}. \end{align*}Since n is a multiple of 7, the smallest positive integer n such that x2+xy+y2+n is a multiple of 7 is 7.
