Mathematical story

At noon today, I was doing my math summer homework. Writing, unfortunately, I have a problem. I thought about it for a long time, but I couldn't figure out a way. The question is this:

There is a cuboid, the product of the front and the upper two areas is 209 square centimeters, and the length, width and height are prime numbers. Find its volume.

I saw it and thought: this question is really difficult! Only knowing the product of two surface areas, the volume must also know the length, width and height, but there is no hint at all. How does this start?

Just as I was scratching my head, a colleague of my mother came. He taught me to use the idea of equation to solve it first, but I am not very familiar with this method of equation. So, he taught me another way: list the numbers first, and then exclude them one by one. First, we listed a lot of numbers according to the requirements of the topic, such as: 3, 5, 7, 1 1, and then we began to exclude them, and then we found that only1and 19 were left. At this time, I thought: one of these two numbers is the length of the common side of the front of the cuboid in the question; One is the front of the cuboid, divided by the sum of the lengths of the other side (both lengths are prime numbers). So, I began to tell which number these two numbers were.

Finally I got the result, which is 374 cubic centimeters. My formula is: 209 =119 = 2+171× 2×17 = 374 (cubic centimeter).

Later, I checked this problem with what I learned this semester: prime factor decomposition, and the results are exactly the same.

I am happier than anyone to solve this problem. I also understand the truth that mathematics is full of mysteries, waiting for us to explore.