2025/030) Show that $12| (a-b)(b-c)(c-a)(a-d)(b-d)(c-d)$ for integers a,b,c,d.

Because of  symmetry a,b,c,d are interchangeable.

 For it to be a multiple of 12 we need to show that it is a multiple of 3 and 4 . This is so because 3 and 4 are co-primes.

 Let us show that it is multiple of 3. As there are 4 numbers a,b,c,d all 4 numbers divided by 3 cannot have 4 differences, So two of them as a and b have same remainder. then (a-b) i divisible by 3

Now let us show that it is is multiple of 4

If at least 3 of them say a,b,c are  even or odd then (a-b) and (b-c) both are even so multiple if product of 4.

If two say a and b are even and other 2 c and d are odd then (a-b) and (c-d) both are even and product is multiple of 4.

As product is multiple of 4 and 3 so it is multiple of 12.