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# Product of 3 consecutive numbers

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Intern
Joined: 27 Oct 2016
Posts: 29
Concentration: Healthcare, Operations
GMAT 1: 710 Q47 V40
GPA: 3.4
WE: Operations (Health Care)
Product of 3 consecutive numbers  [#permalink]

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03 Aug 2017, 13:01
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?
Manhattan Prep Instructor
Joined: 04 Dec 2015
Posts: 729
GMAT 1: 790 Q51 V49
GRE 1: Q170 V170
Re: Product of 3 consecutive numbers  [#permalink]

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03 Aug 2017, 13:22
1
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:

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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Joined: 24 Jun 2008
Posts: 1518
Re: Product of 3 consecutive numbers  [#permalink]

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04 Aug 2017, 05:04
1
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, 05:04
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# Product of 3 consecutive numbers

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