Example 3 Find all positive integers $a, b (a \leqslant b)$, such that
$$a b=300+7[a, b]+5(a, b) .$$
Let $[a, b]=x,(a, b)=y$, by property 8 we know $a b=x y$, thus, (1) becomes
$$x y=300+7 x+5 y$$

which is $(x-5)(y-7)=5 \times 67$.

Since $[a, b] \geqslant(a, b)$, hence $x \geqslant y$, thus $x-5>y-7$, there are only the following two cases.

Case one: $x-5=67$ and $y-7=5$; in this case, $x=72, y=12$, thus, we can set $a=12 n, b=12 m,(m, n)=1$, and have $(12 n)(12 m)=a b=x y=12 \times 72$, combining with $a \leqslant b$, it can only be $(m, n)=(1,6)$ or $(2,3)$, corresponding to $(a, b)=(12,72)$ or $(24, 36)$.

Case two: $x-5=335$ and $y-7=1$; correspondingly, $x=340, y=8$, but $y=(a, b)$ is a factor of $x=[a, b]$, and $8 \nmid 340$, so there is no solution in this case.

In summary, the pairs $(a, b)$ that meet the conditions are $(12,72)$ or $(24,36)$.