Example 3 (IMO-25 Problem) Find positive integers $a, b$ such that:
(1) $ab(a+b)$ is not divisible by 7;
(2) $(a+b)^{7}-a^{7}-b^{7}$ is divisible by $7^{7}$.
Verify your answer.
We have
$$
(a+b)^{7}-a^{7}-b^{7}=7 a b(a+b)\left(a^{2}+a b+b^{2}\right)^{2} \text {. }
$$

From (1) and (2), we know that $7^{7} \mid 7 a b(a+b)\left(a^{2}+a b+b^{2}\right)^{2}$, which means $7^{3} \mid a^{2}+a b+b^{2}$.
Next, we need to find the positive integers $a, b$ that satisfy
$$
(a+b)^{2}-a b=a^{2}+a b+b^{2}=7^{3}=343
$$

Since $0<a b \leqslant \frac{1}{4}(a+b)^{2}$, we get the estimation from (1)
$$
343 \leqslant(a+b)^{2} \leqslant 457 \text {, }
$$

which gives $19 \leqslant a+b \leqslant 21$.
Let $a+b=19$, from (1) we get $a b=18$, solving these equations yields $a=18, b=1$. It is easy to verify that this is a pair of positive integers that satisfy (1) and (2).