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The function ∆(m) is defined for all positive integers m as

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The function ∆(m) is defined for all positive integers m as [#permalink] New post 04 Apr 2018, 12:41
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The function ∆(m) is defined for all positive integers m as the product of m + 4, m + 5, and m + 6. If n is a positive integer, then ∆(n) must be divisible by which one of the following numbers?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 11
[Reveal] Spoiler: OA

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Re: The function ∆(m) is defined for all positive integers m as [#permalink] New post 04 Apr 2018, 21:27
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Firstly, let's understand this function. We're supposed to add 4, 5, and 6 to whatever integer is plugged in, and then multiply the 3 results. It's very useful to know that in any set of, say, 5 consecutive integers, exactly one of them must be divisible by 5. Try it out. Similarly, in the same set of 5 consecutive integers, 1 or 2 of them must be divisible by 4. If the first one is divisible by 4, then the last must also be divisible by 4. If one of the numbers in the middle is divisible by 4, then it'll be the only one. In the same set, 2 or 3 of the numbers will be even. You get the idea.

In this set of 3 consecutive integers, we know that one of them is divisible by 3, and 1 or 2 must be divisible by 2. So if we multiply them all, the result must be divisible by both 3 and 2, or in other words, it's divisible by 6. Thus the answer is C.
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Re: The function ∆(m) is defined for all positive integers m as [#permalink] New post 14 Dec 2018, 17:03
SherpaPrep wrote:
Firstly, let's understand this function. We're supposed to add 4, 5, and 6 to whatever integer is plugged in, and then multiply the 3 results. It's very useful to know that in any set of, say, 5 consecutive integers, exactly one of them must be divisible by 5.


That is a statement I agree with and that I have tested. But how are we supposed to know that in the exam? In other words, what is the deeper mathematical explanation or law that this comes from? If I don't understand, then I might as well forget it now because I will forget it later.
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Re: The function ∆(m) is defined for all positive integers m as [#permalink] New post 15 Dec 2018, 01:14
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No it is the divisibility and consecutive rule and is critical to take into account.

It will permit you to solve a question faster instead to pick numbers and waste your time.

What is important to note is an exam such as GRE (and also GMAT or all the exams which involve math) does not have the equation one rule= I solve this question. Instead, you do have a combination of rules to address a problem. I mean it is like you have to repair a car: it has some mechanic parts for which you do have specific tools, and it also has electronic components for which you do have other tools. Overall, you repair the car. A question is your car to fix: you could have the need of this tool, of that tool or they combined.

I hope this example helps you to get it :)

Regards
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Re: The function ∆(m) is defined for all positive integers m as [#permalink] New post 15 Dec 2018, 17:19
Carcass wrote:
No it is the divisibility and consecutive rule and is critical to take into account.

It will permit you to solve a question faster instead to pick numbers and waste your time.

What is important to note is an exam such as GRE (and also GMAT or all the exams which involve math) does not have the equation one rule= I solve this question. Instead, you do have a combination of rules to address a problem. I mean it is like you have to repair a car: it has some mechanic parts for which you do have specific tools, and it also has electronic components for which you do have other tools. Overall, you repair the car. A question is your car to fix: you could have the need of this tool, of that tool or they combined.

I hope this example helps you to get it :)

Regards


Yeah, I just thought there might be some mathematical law behind it. I tried some examples and saw that is always true. So I pretty much just had to try different things and be creative to get to the solution.
Re: The function ∆(m) is defined for all positive integers m as   [#permalink] 15 Dec 2018, 17:19
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