4. (5 points) Given $a+b=100$, if $a$ divided by 3 leaves a remainder of $2$, and $b$ divided by 7 leaves a remainder of 5, then the maximum value of $a \times b$ is


【Analysis】To maximize $a \times b$, it is necessary to make the difference between $a$ and $b$ as small as possible, while their sum is constant, which can narrow down the range and find the maximum value.

【Solution】Solution: According to the analysis, $a$ divided by 3 leaves a remainder of $2$, and $b$ divided by 7 leaves a remainder of 5. We can set $a=3m+2, b=7n+5$,

Also, since $a+b=100$, with the sum being constant, the smaller the difference, the larger the product, so we get $a=53, b=47$, $a \times b=53 \times 47=2491$, at this point $a \times b$ is maximized. Therefore, the answer is: 2491.