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

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The function f(m) is defined for all positive integers m as [#permalink] New post 26 Jan 2011, 13:00
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The function f(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 f(n) must be divisible by which one of the following numbers?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 11
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Re: Defined Functions [#permalink] New post 26 Jan 2011, 13:38
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PASSINGGMAT wrote:
The function f(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 f(n) must be divisible by which one of the following numbers?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 11

Can you explain how to answer this please. Thank you.


Given: f(n)=(n+4)(n+5)(n+6), where n is a positive integer. Question: f(n) must be divisible by which one of the following numbers.

Now, (n+4)(n+5)(n+6) is the product of 3 consecutive integers so out of them one is definitely divisible by 3 and at least one is divisible by 2, so f(n) must be divisible by 2*3=6.

Answer: C.

Generally out of ANY k consecutive integers one is always divisible by k and at least one by k-1, k-2, ... For example out of ANY 5 consecutive integers there is one which is divisible by 5, and at least one which is divisible by 4, 3, and 2. That's because an integer divided by an integer k can give a remainder of: 0 (when it's divisible by k), 1, 2, ..., or k-1 (total of k different remainders from 0 to k-1), so out of k consecutive integers there definitely will be one which gives a reminder of zero, so divisible by k.

Which give us the following property: the product of k consecutive integers is always divisible by k!, so by k too. For example: given k=4 consecutive integers \{3,4,5,6\} --> the product of 3*4*5*6 is 360, which is divisible by 4!=24.

If we apply this property to the original question we'll have that the product of given 3 consecutive integers (n+4)(n+5)(n+6) must be divisible by 3!=6.

Hope it's clear.
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Re: Defined Functions [#permalink] New post 27 Jan 2011, 21:45
awesome explanation
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Re: Defined Functions [#permalink] New post 29 Jan 2011, 11:59
Ye, Bunuel - u rocked it. +1.
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Re: Defined Functions [#permalink] New post 08 Mar 2011, 19:14
Product of Any three consecutive integers must be divisible by 6.
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Re: The function f(m) is defined for all positive integers m as [#permalink] New post 03 Oct 2013, 18:46
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Re: The function f(m) is defined for all positive integers m as   [#permalink] 03 Oct 2013, 18:46
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