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The function ∆(m) is defined for all positive integers m as
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04 Apr 2018, 12:41
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
(A) 4
(B) 5
(C) 6
(D) 7
(E) 11
Sherpa Prep Representative
Joined: 15 Jan 2018
Posts: 147
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Re: The function ∆(m) is defined for all positive integers m as
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04 Apr 2018, 21:27
04 Apr 2018, 21:27
3
Expert Reply
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.
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.
Re: The function ∆(m) is defined for all positive integers m as
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14 Dec 2018, 17:03
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.
