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Product of 3 consecutive numbers

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Product of 3 consecutive numbers [#permalink]

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New post 03 Aug 2017, 12:01
So I started wondering if this rule applies to all numbers.
For example, product of 4 consecute numbers divisible by 4, product of 5 consecutive numbers divisible by 5 etc.

Why is the only focussed for 3? Does this apply to all numbers. I've tried using examples to disprove it for other numbers but maybe I havent thought of all scenarios.

Does this work for all numbers?

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Re: Product of 3 consecutive numbers [#permalink]

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New post 03 Aug 2017, 12:22
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Yes, this is true for all numbers!

Think about the consecutive numbers as cars on a train. The number line is the train track. If the train is three cars long, it can travel along the track like this:

Image

No matter what, one of the cars will always be at a stop that's a multiple of 3. So, when you multiply the values of the three cars together, the result will always be a multiple of 3 as well.

The same is true if you have a car of length 4 (one car has to always be at a stop that's a multiple of 4), etc - draw it out and convince yourself :)
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Re: Product of 3 consecutive numbers [#permalink]

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New post 04 Aug 2017, 04:04
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You can even take this further. As you can see in Chelsea's example above, among three consecutive integers you not only find a multiple of 3 somewhere -- you also must have at least one multiple of 2. So the product of three consecutive integers will always be divisible by 3*2, or in other words, by 3! (three factorial).

If instead you look at four consecutive numbers, you'll always have exactly one multiple of 4, at least one multiple of 3, and another number which is a multiple of 2, so the product of four consecutive integers is always divisible by 4*3*2, or in other words by 4!

And that's a general rule: the product of n consecutive integers is always divisible by n!
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Re: Product of 3 consecutive numbers   [#permalink] 04 Aug 2017, 04:04
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Product of 3 consecutive numbers

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