neerajkumar1_1 wrote:If an even number n is not divisible by 3 or 4, then what must (n + 6)(n + 8)(n + 10) be divisible by?
I. 24
II. 32
III. 96
A. None
B. I only
C. II only
D. I and II only
E. I, II, and III

OA:E

n is an even number so n+6 will be an evn number n+8 will also be even number n+10 will also be even number
if you look at it carefully its 3 continious even number .and n should not be divisible by 3 or 4 so let us take one smallest value of n i.e 0 or for sake 2 i.e
if n=2

8*10*12=2*2*2*2*5*2*2*3 so it will contain 24 ,32 ,96 all of thm so i guess answer will be E
you cant take 0 as 0 isdivsible by 4 i.e 0/4=0
een able to get the answer and I have done it just over 2 mins. I solved the question by realizing 2, 10 and 14 will not be divisble by 2 & 3 and then factorizing (n + 6)(n + 8)(n + 10) to find the common factors in 24,32 and 96.

Now when i was looking at the explanation i found it a better way to think:
So here is the explanation:

If n is an even number not divisible by 4, then n + 8 will also be an even number not divisible by 4.
However, n + 6 and n + 10 will both be even numbers that are divisible by 4.
Furthermore, n + 6 and n + 9 wouldn't be divisible by 3 either, but either n + 8 or n +10 must be divisible be 3
because at least one of three consecutive integers must be divisible by 3.
All together, we find that the product (n + 6)(n + 8)(n + 10) will be divisible by 2 × 4 × 4 × 3.


I find the method pretty good, but I am unable to understand the last step in which it mentions:
All together, we find that the product (n + 6)(n + 8)(n + 10) will be divisible by 2 × 4 × 4 × 3.


Here I get 4 x 4 x 3..... but i dont understand how according to their logic they also add 2 in the must be divisors.

Please let me know your views...

Thanks...

neerajkumar1_1 wrote:If an even number n is not divisible by 3 or 4, then what must (n + 6)(n + 8)(n + 10) be divisible by?
I. 24
II. 32
III. 96
A. None
B. I only
C. II only
D. I and II only
E. I, II, and III

OA:E

I have been able to get the answer and I have done it just over 2 mins. I solved the question by realizing 2, 10 and 14 will not be divisble by 2 & 3 and then factorizing (n + 6)(n + 8)(n + 10) to find the common factors in 24,32 and 96.

Now when i was looking at the explanation i found it a better way to think:
So here is the explanation:

If n is an even number not divisible by 4, then n + 8 will also be an even number not divisible by 4.
However, n + 6 and n + 10 will both be even numbers that are divisible by 4.
Furthermore, n + 6 and n + 9 wouldn't be divisible by 3 either, but either n + 8 or n +10 must be divisible be 3
because at least one of three consecutive integers must be divisible by 3.
All together, we find that the product (n + 6)(n + 8)(n + 10) will be divisible by 2 × 4 × 4 × 3.


I find the method pretty good, but I am unable to understand the last step in which it mentions:
All together, we find that the product (n + 6)(n + 8)(n + 10) will be divisible by 2 × 4 × 4 × 3.


Here I get 4 x 4 x 3..... but i dont understand how according to their logic they also add 2 in the must be divisors.

Please let me know your views...

Thanks...

raz1024 wrote:Hey Neeraj,

While it is well and good to search for thorough algebraic explanations when you have the time to do so, it is not very helpful when you have only 2 minutes average to answer a question.

This particular question is one solved much more easily with testing numbers. You are searching for what the product MUST BE divisible by. The best strategy, therefore, is to choose the lowest possible value of n, so you have the fewest possible number of divisors of (n+6)(n+8)(n+10).

By generating the fewest possible number of divisors, you are being more restrictive, which means you will find out how many of the three given options (24, 23, 96) MUST ALWAYS be factors of the product.

In this case, the smallest possible value of n is 2. In that case, the product is (2+6)(2+8)(2+10) = 960.

960 is divisible by 24, 32, and 96; therefore, the answer is E.

This might not be the most complete approach to solving all of the algebraic properties to this question, but your goal is just to find the correct answer. That is all you should be concerned about.

Hope this helps!

jube wrote:
I have a doubt about this approach. While (2+6)(2+8)(2+10) is divisble by all the nos., how do we know for sure whether another no. say 5 or 10 or 62 or any other no. which satisfies the condition will also necessarily be divisble by all the nos.

thanks!

kvcpk wrote:I was finally able to find the algebraic solution for this problem.. Though not important from GMAT perspective.. Its good to know the approach..
Here it comes..

Let the even number be 2k.

We need to find what (n + 6)(n + 8)(n + 10) is divisible by:
Substituting n=2k, we get,
(2k + 6)(2k + 8)(2k + 10)
= 8(k+ 3)(k+ 4)(k+ 5)

Given that 2k is not divisible by 3 or 4.
So 2k should be of the form:
3a +1 or
3a + 2 or
4b + 1 or
4b + 2 or
4b + 3

jube wrote: quick question about the above - most likely I'm missing something so will be awesome if you could clarify.

If we take 2k to be 3n + 1 then won't 4 also figure in there? similarly, if we take it to be of the form 3n + 2 then won't 8 also figure in the list?

thanks!

neerajkumar1_1 wrote:
If n is an even number not divisible by 4, then n + 8 will also be an even number not divisible by 4.
However, n + 6 and n + 10 will both be even numbers that are divisible by 4.
Furthermore, n + 6 and n + 9 wouldn't be divisible by 3 either, but either n + 8 or n +10 must be divisible be 3
because at least one of three consecutive integers must be divisible by 3.
All together, we find that the product (n + 6)(n + 8)(n + 10) will be divisible by 2 × 4 × 4 × 3.


I find the method pretty good, but I am unable to understand the last step in which it mentions:
All together, we find that the product (n + 6)(n + 8)(n + 10) will be divisible by 2 × 4 × 4 × 3.


Here I get 4 x 4 x 3..... but i dont understand how according to their logic they also add 2 in the must be divisors.

Please let me know your views...

Thanks...

raz1024 wrote:Hey Neeraj,

While it is well and good to search for thorough algebraic explanations when you have the time to do so, it is not very helpful when you have only 2 minutes average to answer a question.

This particular question is one solved much more easily with testing numbers. You are searching for what the product MUST BE divisible by. The best strategy, therefore, is to choose the lowest possible value of n, so you have the fewest possible number of divisors of (n+6)(n+8)(n+10).

By generating the fewest possible number of divisors, you are being more restrictive, which means you will find out how many of the three given options (24, 23, 96) MUST ALWAYS be factors of the product.

In this case, the smallest possible value of n is 2. In that case, the product is (2+6)(2+8)(2+10) = 960.

960 is divisible by 24, 32, and 96; therefore, the answer is E.

This might not be the most complete approach to solving all of the algebraic properties to this question, but your goal is just to find the correct answer. That is all you should be concerned about.

Hope this helps!