摘要：However, since we only have the digits 0, 2, 3, 6, and 7, let's re-evaluate:
用0、2、3、6、7组成三位数乘两位数的乘法算式，乘积最大是多少？最小是多少？
为了找到由数字0、2、3、6、7组成的三位数乘两位数的乘法算式中乘积的最大值和最小值，我们需要考虑以下几点：
### 最大乘积
1. **选择最大的 digits for the largest number:**
- The largest digit available is 7.
- The next largest digit is 6.
2. **Form the two numbers:**
- To maximize the product, we should place the largest digits in the highest places (hundreds and tens).
- Therefore, the three-digit number should be formed with the remaining largest digits after forming the two-digit number.
Let's start by trying different combinations:
- If the two-digit number is $76$:
- The remaining digits are $5$ (since we don't have a 4 or 9), so the three-digit number could be $ \text{3} $ (the next largest digit).
However, since we only have the digits 0, 2, 3, 6, and 7, let's re-evaluate:
- If the two-digit number is $76$:
- The remaining digits are $3, 2, 0$. The largest possible three-digit number is $320$.
Now calculate the product:
$$ 76 \times 320 = 24320 $$
- If the two-digit number is $73$:
- The remaining digits are $6, 2, 0$. The largest possible three-digit number is $620$.
Now calculate the product:
$$ 73 \times 620 = 45260 $$
- If the two-digit number is $72$:
- The remaining digits are $6, 3, 0$. The largest possible three-digit number is $630$.
Now calculate the product:
$$ 72 \times 630 = 45360 $$
- If the two-digit number is $67$:
- The remaining digits are $3, 2, 0$. The largest possible three-digit number is $320$.
Now calculate the product:
$$ 67 \times 320 = 21440 $$
From these calculations, the maximum product is:
$$ 72 \times 630 = 45360 $$
### Minimum Product
To find the minimum product, we need to form the smallest possible numbers using all five digits.
1. **Choose the smallest digits for the smallest number:**
- The smallest non-zero digit available is 2.
- The next smallest digit is 3.
2. **Form the two numbers:**
- To minimize the product, we should place the smallest digits in the highest places (hundreds and tens).
- Therefore, the three-digit number should be formed with the remaining smallest digits after forming the two-digit number.
Let's try different combinations:
- If the two-digit number is $23$:
- The remaining digits are $7, 6, 0$. The smallest possible three-digit number is $607$.
Now calculate the product:
$$ 23 \times 607 = 13961 $$
- If the two-digit number is $26$:
- The remaining digits are $7, 3, 0$. The smallest possible three-digit number is $307$.
Now calculate the product:
$$ 26 \times 307 = 7982 $$
- If the two-digit number is $27$:
- The remaining digits are $6, 3, 0$. The smallest possible three-digit number is $306$.
Now calculate the product:
$$ 27 \times 306 = 8262 $$
- If the two-digit number is $32$:
- The remaining digits are $7, 6, 0$. The smallest possible three-digit number is $607$.
Now calculate the product:
$$ 32 \times 607 = 19424 $$
- If the two-digit number is $36$:
- The remaining digits are $7, 2, 0$. The smallest possible three-digit number is $207$.
Now calculate the product:
$$ 36 \times 207 = 7452 $$
- If the two-digit number is $37$:
- The remaining digits are $6, 2, 0$. The smallest possible three-digit number is $206$.
Now calculate the product:
$$ 37 \times 206 = 7622 $$
- If the two-digit number is $62$:
- The remaining digits are $7, 3, 0$. The smallest possible three-digit number is $307$.
Now calculate the product:
$$ 62 \times 307 = 19034 $$
- If the two-digit number is $63$:
- The remaining digits are $7, 2, 0$. The smallest possible three-digit number is $207$.
Now calculate the product:
$$ 63 \times 207 = 13041 $$
- If the two-digit number is $67$:
- The remaining digits are $3, 2, 0$. The smallest possible three-digit number is $203$.
Now calculate the product:
$$ 67 \times 203 = 13561 $$
- If the two-digit number is $72$:
- The remaining digits are $6, 3, 0$. The smallest possible three-digit number is $306$.
Now calculate the product:
$$ 72 \times 306 = 21912 $$
- If the two-digit number is $73$:
- The remaining digits are $6, 2, 0$. The smallest possible three-digit number is $206$.
Now calculate the product:
$$ 73 \times 206 = 15038 $$
- If the two-digit number is $76$:
- The remaining digits are $3, 2, 0$. The smallest possible three-digit number is $203$.
Now calculate the product:
$$ 76 \times 203 = 15428 $$
From these calculations, the minimum product is:
$$ 26 \times 307 = 7982 $$
Thus, the final answers are:
$$
\boxed{45360}
$$
for the maximum product and
$$
\boxed{7982}
$$ for the minimum product.
来源：中华科学之家一点号