13. (15 points) Write down all positive integers $a$ and $b$ that satisfy the following three conditions:
(1) $a \leqslant b$;
(2) $a+b$ is a three-digit number, and the three digits are in arithmetic progression;
(3) $a \times b$ is a five-digit number, and all five digits are the same.
【Analysis】First, decompose the smallest five-digit number 11111 with all digits the same, then discuss the values of $a$ and $b$ based on the conditions they satisfy.

【Solution】Solution: $11111=41 \times 271$
(1) $a=271, b$ is the smallest as $41 \times 7=287$, and the largest as $41 \times 9=369$

Upon verification, it does not satisfy the conditions.
(2) $b=271, a$ could be $41,82,123,164,205,246$

Upon verification, $a=41,164$,
(3) $b=542, a$ could be $41,82,123 \cdots 451$

Upon verification, $a=82$,
(4) $b=813, a$ could be $41,82,123,164$

Upon verification, $a=123$
In summary, the positive integers $a, b$ that meet the requirements are:
$(41,271),(164,271),(82,542),(123,813)$