The product of four consecutive numbers

I recently came across something new (to me) but not (see below) new to google.

Consider four consecutive numbers, say 2,3,4 and 5. What is their product?- 120; or take 3,4,5,6…360. In both cases, the product is a square number less 1: 121, 11^2, and 361, 19^2.

It is fairly straightforward by algebra to prove that this result is always the case. I say straightforward, it is, if you know the answer you are looking for and can work back, factorising a quartic equation. How it was discovered, I don't know.

I like to solve such problems geometrically or intuitively, but have failed with this problem.

It is perhaps somehow related to the fact that, taking the first example, the product of the outside 2 is 1 less than 11, whilst the product of the inner two is 1 more- so a rectangle 10*12, ie with area 121; and this applies in all cases.

Or, perhaps it is somehow related to the fact that the products are always divisible by 24, since four numbers must have two even numbers and at least one number divisible by three: and 24 is one less than a square.

Or, it is not due to either of these. When I googled the problem, I found lots of pages of similar analysis, but nothing by way of illustration. Also, of course, I found out that eminent mathematicians like Euler had solved far more complex examples. The sums of n numbers are always 1 less than a square; and the product of n numbers is always divisible by n! ( n factorial)-as said, in the case of 4, by 24.

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