a
\(\cos \theta=\frac{5}{9}\)

\(\frac{\vec{a} \cdot \vec{b}}{a b}=\frac{5}{9}\)

\(\mid \vec{a}+\cdots(1)\)

\(a^2+b^2+2 \vec{a} \cdot \vec{b}=2 a^2+2 b^2-4 \vec{a} \cdot \vec{b}\)

\(6 \vec{a} \cdot \vec{b}=a^2+b^2\)

\(6 \times \frac{5}{9} a b=a^2+b^2\)

\(\frac{10}{3} a b=a^2+b^2 \quad \& \quad a=n b\)

\(\frac{10}{3} n b^2=n^2 b^2+b^2\)

\(3 n^2-10 n+3=0\)

\(n=\frac{1}{3} \text { and } n=3\)

\(\text { integer value } n=3\)