4. (25th IMO Problem) Find a pair of positive integers $a, b$ satisfying:
(1) $ab(a+b)$ is not divisible by 7.
(2) $(a+b)^{7}-a^{7}-b^{7}$ is divisible by $7^{7}$.
4. From $7^{7} \mid\left[(a+b)^{7}-a^{7}-b^{7}\right]=7 a b(a+b)\left(a^{2}+a b+b^{2}\right)^{2}$ and $7 a b(a+b)$, we know $7^{3} \mid a^{2}+a b+b^{2}$, i.e., $a^{2}+a b+b^{2}=7^{3} t$ (where $t$ is a positive integer). When $t=1$, $a^{2}+a b+b^{2}=343$. Since $(a+b)^{2}-a b=a^{2}+a b+b^{2} \geqslant 3 a b$, we have $3 a b \leqslant 343, a b \leqslant 114$. Thus, $343<(a+b)^{2} \leqslant 343+114=457, 19 \leqslant a+b \leqslant 21$. Let $a+b=19$, then $a b=(a+b)^{2}-343=18, a=18, b=1$. Upon verification, $(a, b)=(18,1)$ satisfies the conditions of the problem.