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

Question

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

Bunuel · Accepted Answer

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.

gmat1220 · Answer

Product of Any three consecutive integers must be divisible by 6.

EMPOWERgmatRichC · Answer

Hi All,

This question is based on a couple of Number Properties; as such, you can also TEST VALUES to get to the solution.

We're told that the f(M) = (M+4)(M+5)(M+6) for all POSITIVE integers. We're asked which of the following numbers MUST divide into f(N).

IF....
N = 1
f(1) = (5)(6)(7)

At this point, you can either multiply out the numbers and check the 5 answer choices against that product OR prime factor the f(1)....

(5)(6)(7) = (5)(2)(3)(7)

Of the 5 answer choices, only 2 of them divide into this product (Answers B and C; 5 and 6).

From here, we should look to try to eliminate one of the options. It's actually not that hard....

IF....
N = 2
f(2) = (6)(7)(8)

Looking at this, we can see that 6 IS a factor while 5 is NOT.

Final Answer:  C

GMAT assassins aren't born, they're made,
Rich

pnh0505 · Answer

I tried the sum with f(13), which is equal to 17 X 18 X 19 (this number isn't divisible by 6)?

EMPOWERgmatRichC · Answer

Hi pnh0505,

You are correct that when you use M=13, then f(13) = (17)(18)(19). However, that product IS divisible by 6 - because 18 is divisible by 6.

GMAT assassins aren't born, they're made,
Rich

KrishnakumarKA1 · Answer

In three consecutive integers we can say that one of it will be an even number and one will a multiple of three.
To the product will be definitely the product of 6.
Option C

ManifestDreamMBA · Answer

f(m) = (m + 4) * (m + 5) * (m + 6)
Let k = m+5
f(m) = (k-1) * k * (k+1) , which is nothing but 3 consecutive integers

n consecutive integers are divisible by n!

This means f(m) or f(n) will be divisible by 3! = 6

Answer C

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

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