explain product of consec. integers and divisibility

Hey Gurpinder,

Good question - I find that, a lot of times, the need for precise definitions may detract from the simplicity that you can get from proving the rule to yourself. Whenever you're confused on a definition, see if you can make it make sense by testing it with small numbers (which is a big-picture number properties strategy, too - try to establish patterns with small numbers!).

Here's what they're saying:

2 numbers:

If you plot them on a number line you can see that every second number is even (which means that it's divisible by 2):

1, 2, 3, 4, 5, 6, 7, 8....

Pick any two consecutive numbers from that list and you're guaranteed that one will be even, so if you multiply them together you'll definitely get an even number.

3 numbers:

Try the same thing, plotting them on a number line - every third number is a multiple of 3:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12...

If you pick any three consecutive numbers, one of them will be one of those multiples of 3:

1, 2, 3

2, 3, 4

3, 4, 5

4, 5, 6

That means that if you multiply any three consecutive integers, you'll end up with a multiple of three.

This pattern will extrapolate such that "K" consecutive integers will be divisible by "K". And it's K!, too, because as you get smaller than K, you're including all of those factors, too. Say you took 5 consecutive integers:

16 * 17 * 18 * 19 * 20

20 is a multiple of 5, so it's divisible by 5
20 is also a multiple of 4 so it's divisible by 4 (so is 16, so we're double-covered)
18 is a multiple of 3, so we have that
16, 18, and 20 are all multiples of 2, so we have that

Essentially, five consecutive integers contains four, three, and two consecutive integers, so all of those smaller values are included.

I hope that helps... As someone who has written some math books I know how tough it is to be a combination of precise, concise, and helpful all at the same time! When in doubt, try to prove these rules to yourself, and you'll usually not forget them once you understand why they work.

Thanks Brian,

You are very helpful!!!!!

I don't have this book... and never realized this concept... though have solved a couple of questions based on this...