A big mistake

The question is like this. For any integer a, B, c, d, the expression is:

(A-B) (A-C) (A-D) (B-C) (B-D)

Value can be divisible by 12.

My idea is correct. If four integers are divided by three, then the remainder of at least two numbers is the same. Then the difference between the two numbers is

It can be divisible by 3, so that the 3 factor can be found in the expression. For example, it can be proved that there are two 2

So that the entire expression can be divisible by 12.

Although the idea is correct, the specific method for finding a factor is wrong. To prove that a factor 3 exists, I just think that we can find the package

An expression with four numbers can be used. For example, if the following expression is used, I think there are three factors:

 

(A-B) (A-C) (A-D)

 

Because it contains all four different numbers, but unfortunately this is wrong. Only the four numbers with different numbers are two or two subtracted.

When a factor exists, it can ensure that there are 3 factors in the expression. For example:

 

A = 10, B = 2, c = 5, D = 6

 

So (A-B) (A-C) (A-D) There is no 3 factor. But the expression of the 4 Number 2, 2 subtraction Product

 

(A-B) (A-C) (A-D) (B-C) (B-D)

 

There are three factors.

When this error is found, immediately correct it and try to find the correct proof, but it is a pity that two factors are not available.
Yes. After thinking for a long time, I finally gave up and looked at the correct answer. The result was not a two-factor, but a four-factor.
Factor. It turns out that it is very easy to use the Pigeon nest principle again. If the four numbers are divided by four and there is the same remainder, then 2 and 2 are subtracted.
There is a factor 4 in the expression of product. This is easy. The key is the next step, that is, the same remainder does not exist.
The four remainder numbers are 0, 1, 2, and 3 respectively, and the four numbers take one of them. If they are not general, we can assume that A, B, C, and D correspond to the four remainder numbers.
Then (a-c) (B-d) contains two factors, so that the original proposition is proved.

I learned a lot from the mistakes I made in this question and the answers I got. I have a deep understanding of them.
The actual application is of great help.