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If n is an integer greater than 6, which of the following mu

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If n is an integer greater than 6, which of the following mu [#permalink] New post 20 Sep 2012, 16:26
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Re: If n is an integer greater than 6, which of the following mu [#permalink] New post 31 Mar 2013, 04:55
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Since this is a must be true question it must be true for any n and as we know any number is divisible by 3 if the sum of its digits is divisible by 3, so if we add up the digits or terms in the options we get the answer

(A) n + (n + 1) + (n – 4) = 3n - 3 ---- divisible by 3 for any n
(B) n + (n + 2) + (n – 1) = 3n + 1
(C) n + (n + 3) + (n – 5) = 3n - 2
(D) n + (n + 4) + (n – 2) = 3n + 2
(E) n + (n + 5) + (n – 6) = 3n - 1
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Last edited by prasun9 on 08 Oct 2013, 04:38, edited 2 times in total.
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Re: If n is an integer greater than 6, which of the following mu [#permalink] New post 03 Jul 2013, 20:51
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Which method is better in questions like these to apply on the test? to pick the numbers or to apply concept of divisibility by 3 ( Remainder 0,1,2)



In my opinion, using logic is almost always better than plugging in. Plugging in is full of possibilities of making mistakes - incorrect calculation, not considering all possibilities, getting lost in the options etc.

Logic is far cleaner.

I know that talking about positive integers, in any set of 3 consecutive positive integers, one integer will be divisible by 3 and the other 2 will not be.

So I am looking for 3 consecutive positive integers e.g. (n-1)n(n+1)

Note that (n-4) is equivalent to (n-1) since if (n-4) is divisible by 3, so is (n-1). If (n-4) is not divisible by 3, neither is (n-1) (because the difference between these two integers is 3)
Hence (A) is equivalent to 3 consecutive integers.
Answer (A)

On same lines, note that n is equivalent to (n-3), (n + 3), (n + 6) etc.
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Re: If n is an integer greater than 6, which of the following mu [#permalink] New post 21 Sep 2012, 00:13
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If n is an integer greater than 6, which of the following must be divisible by 3 ?

(A) n(n + 1)(n – 4)
(B) n(n + 2)(n – 1)
(C) n(n + 3)(n – 5)
(D) n(n + 4)(n – 2)
(E) n(n + 5)(n – 6)


Since 3 is a prime number then in order the product to be divisible by 3 either of the multiples must be divisible by 3. Now, to guarantee that at least one multiple is divisible by 3, these numbers must have different remainders upon division by 3, meaning that one of them should have the remainder of 1, another the reminder of 2 and the third one the remainder of 0, so be divisible by 3.

For option A: n and n+1 have different remainder upon division by 3. As for n-4, it will have the same remainder as (n-4)+3=n-1, so also different than the remainders of the previous two numbers.

Answer: A.

Similar question to practice: if-x-is-an-integer-then-x-x-1-x-k-must-be-evenly-divisible-126853.html

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Re: If n is an integer greater than 6, which of the following mu [#permalink] New post 21 Sep 2012, 05:54
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I attacked the problem in this was, tell me if pron of errors

n is an integers, so we can choose 7, 8, 9 and so on. Now, a number divisible by 3 the sum of number MUST be divisible by 3.

1) n (n+1)(n-4) ---> 8 * 9 * 4 ---> without perform multiplication the SUM of 8 + 9 + 4 = 21 and is divisible by 3 without reminder

The rest of choices do not work if you try.

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Re: If n is an integer greater than 6, which of the following mu [#permalink] New post 24 Sep 2012, 10:39
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Shortcut: In every set of 3 consecutive numbers, ONE of them must be divisible by 3 when we are multiplying each of the digits

1*2*3
2*3*4
3*4*5
4*5*6

(A) n(n + 1)(n – 4)

The easiest is if we have something like n(n+1)(n+2)
We know in this case we DEFINITELY have an expression that is divisible by 3.
n=1 => 1*2*3
n=2 => 2*3*4
n=3 => 3*4*5

All are divisible by 3.

Any expression must pass our 3 consecutive integer test.

n(n + 1)(n – 4)
n=1 => 1*2*-3
n=2 => 2*3*1
n=3 => 3*4*-1

Even if n = 4 we have:
4*5*0 = 0 which is divisible by 3.

If n = 16
16*17*12 is divisible by 3.

So (A) passes all the tests and one of numbers in the expression will be divisible by 3 so the whole expression when multiplied together will be divisible by 3.
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Re: If n is an integer greater than 6, which of the following mu [#permalink] New post 25 Sep 2012, 23:02
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Every three consecutive positive integers will have one value divisible by 3.
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Re: If n is an integer greater than 6, which of the following mu [#permalink] New post 20 Sep 2012, 17:28
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carcass wrote:
If n is an integer greater than 6, which of the following must be divisible by 3 ?

(A) n(n + 1)(n – 4)
(B) n(n + 2)(n – 1)
(C) n(n + 3)(n – 5)
(D) n(n + 4)(n – 2)
(E) n(n + 5)(n – 6)

I like this one.I do not find it discussed earlier on the board.

OA later :)


Pick odd and even numbers, e.g. 7 and 8:

A)
(7)(8)(3) Y ; (8)(9)(5) Y
B)
(7)(9)(6) Y ; (8)(10)(7) N
C)
(7)(10)(2) N; No need to check for 8
D)
(7)(11)(5) N; No need to check for 8
E)
(7)(12)(1) Y; (8)(13)(2) N

Only A works for both cases.
A
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Re: If n is an integer greater than 6, which of the following mu [#permalink] New post 08 Feb 2013, 07:56
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carcass wrote:
If n is an integer greater than 6, which of the following must be divisible by 3 ?

(A) n(n + 1)(n – 4)
(B) n(n + 2)(n – 1)
(C) n(n + 3)(n – 5)
(D) n(n + 4)(n – 2)
(E) n(n + 5)(n – 6)



The best way to tackle these kind of questions is by assuming values..
Since its says, n has to be greater than 6...assume the value of n to be 7....by this u will be able to eliminate options 3 and 4...
Now take the value of n as 8....by this u will be able to eliminate options 2 and 5...
Now we are left with only option 1...and that is the answer..
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Re: If n is an integer greater than 6, which of the following mu [#permalink] New post 03 Jul 2013, 19:58
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We need to check only for 3 values of n because the answer will be the same for every 3 consecutive integers. Actually we need not consider one of those values as it is always a multiple of 3.

So we can take the values of n as 7 and 8 or 10 and 11 or 13 and 14 etc

If we plug in the values 7 and 8 we get the answer once we check A as it satisfies both the values. We need not actually check other choices as they would fail for either 7 or 8.
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Re: If n is an integer greater than 6, which of the following mu [#permalink] New post 08 Oct 2013, 20:25
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khairilthegreat wrote:
Dear Bunuel,

"Now, to guarantee that at least one multiple is divisible by 3, these numbers must have different remainders upon division by 3"

Why is this? Why the numbers must have different reminders?


Any positive number will take one of three forms: 3m, (3m+1) or (3m+2) i.e. either it will be divisible by 3, will leave remainder 1 or will leave remainder 2 when divided by 3. If the number takes the form 3m, the number after it is of the form 3m+1 and the one after it is of the form 3m+2.

If we have 3 consecutive numbers such as a, (a+1), (a+2), we know for sure that at least one of them is divisible by 3 since one of them will be of the form 3m. We don't know which one but one of them will be divisible by 3.

So given numbers such as (n-1)*n*(n+1), we know that the product is divisible by 3.
In the given options, we don't know whether n is divisible by 3 or not. We need to look for the option which has 3 consecutive numbers i.e. in which the terms leave a remainder of 0, 1 and 2 to be able to say that the product will be divisible by 3.

Note a product such as n(n+3)(n+6). When this is divided by 3, we cannot say whether it is divisible or not because all three factors will leave the same remainder, 1.
Say n = 4. Product 4*7*10. All these factors are of the form 3m+1. We don't have any 3m factor here.
So we need the factors to have 3 different remainders so that one of them is of the form 3m.
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Re: If n is an integer greater than 6, which of the following mu [#permalink] New post 10 Oct 2013, 20:46
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Re: If n is an integer greater than 6, which of the following mu [#permalink] New post 22 Sep 2012, 10:06
Bunuel wrote:
carcass wrote:
If n is an integer greater than 6, which of the following must be divisible by 3 ?


Since 3 is a prime number then in order the product to be divisible by 3 either of the multiples must be divisible by 3. Now, to guarantee that at least one multiple is divisible by 3, these numbers must have different remainders upon division by 3, meaning that one of them should have the remainder of 1, another the reminder of 2 and the third one the remainder of 0, so be divisible by 3.

For option A: n and n+1 have different remainder upon division by 3. As for n-4, it will have the same remainder as (n-4)+3=n-1, so also different than the remainders of the previous two numbers.



Hi Bunuel - I didn't quite follow this logic. Could you please elaborate. Thanks.

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Re: If n is an integer greater than 6, which of the following mu [#permalink] New post 24 Sep 2012, 03:06
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Jp27 wrote:
Bunuel wrote:
carcass wrote:
If n is an integer greater than 6, which of the following must be divisible by 3 ?


Since 3 is a prime number then in order the product to be divisible by 3 either of the multiples must be divisible by 3. Now, to guarantee that at least one multiple is divisible by 3, these numbers must have different remainders upon division by 3, meaning that one of them should have the remainder of 1, another the reminder of 2 and the third one the remainder of 0, so be divisible by 3.

For option A: n and n+1 have different remainder upon division by 3. As for n-4, it will have the same remainder as (n-4)+3=n-1, so also different than the remainders of the previous two numbers.



Hi Bunuel - I didn't quite follow this logic. Could you please elaborate. Thanks.

Cheers


Can you please tell me which part didn't you understand?

Meanwhile check this question: if-x-is-an-integer-then-x-x-1-x-k-must-be-evenly-divisible-126853.html It might help.
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Re: If n is an integer greater than 6, which of the following mu [#permalink] New post 24 Sep 2012, 06:56
Bunuel wrote:

Can you please tell me which part didn't you understand?

Meanwhile check this question: if-x-is-an-integer-then-x-x-1-x-k-must-be-evenly-divisible-126853.html It might help.


Thanks for your response. the link really helped. Im able to solve these problems.. thanks :-)

cheers
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Re: If n is an integer greater than 6, which of the following mu [#permalink] New post 25 Sep 2012, 20:36
gmatpill wrote:
Shortcut: In every set of 3 consecutive numbers, ONE of them must be divisible by 3 when we are multiplying each of the digits

1*2*3
2*3*4
3*4*5
4*5*6

(A) n(n + 1)(n – 4)

The easiest is if we have something like n(n+1)(n+2)
We know in this case we DEFINITELY have an expression that is divisible by 3.
n=1 => 1*2*3
n=2 => 2*3*4
n=3 => 3*4*5

All are divisible by 3.

Any expression must pass our 3 consecutive integer test.

n(n + 1)(n – 4)
n=1 => 1*2*-3
n=2 => 2*3*1
n=3 => 3*4*-1

Even if n = 4 we have:
4*5*0 = 0 which is divisible by 3.

If n = 16
16*17*12 is divisible by 3.

So (A) passes all the tests and one of numbers in the expression will be divisible by 3 so the whole expression when multiplied together will be divisible by 3.


Thanks gmatpill - I have a basic question here ->

11 has a reminder 4 on division by 7
12 has a reminder 5 on division by 7
What is the reminder x * y divide by 7? - I can multiply reminders as long as I correct the excess amount... so 4 *5 = 20
20 - 7*2 = 6, which is the reminder when x * y / 7 - SO FAR CORRECT?

Reminder when 99 / 15 give me a reminder of 9
But when i factorize the den into primes = 9 * 11 / 3 * 5 => 9/3 leaves reminder 0 and 11/5 reminder leaves 1
On multiplying the reminders, we get zero. So I should not factorize the denominator unless the denominator factor into the same primes

IS THIS UNDERSTANDING CORRECT? MANY THANKS. :|
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Re: If n is an integer greater than 6, which of the following mu [#permalink] New post 01 Oct 2012, 08:11
I tried plugging in odd and even values for n and got the answer A ...

If we assume n as being either 7 , 8 or 9 we can easily see why A is the correct answer .. Because in every instance , either of the 3 numbers multiplied will be divisible by 3 and the entire product will meet the requirements..

if n =7 , then n-4 is divisible by 3 ...
If n= 8 then n+1 is divisible by 3
If n= 9 , then n is divisible by 3 ...
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Re: If n is an integer greater than 6, which of the following mu [#permalink] New post 30 Dec 2012, 09:14
mbaiseasy wrote:
Every three consecutive positive integers will have one value divisible by 3.


Beautiful solution Irene,

Picturing the number line is a 15 second approach to this problem.
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Re: If n is an integer greater than 6, which of the following mu [#permalink] New post 08 Feb 2013, 06:14
just draw a table for each ans option plug in starting 7,8,9 etc. It should not take more than 75 s.
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Re: If n is an integer greater than 6, which of the following mu [#permalink] New post 25 Feb 2013, 20:28
the most easy way is to pick numbers and plug in each choice. in the test room, we have no time for thinking hard way. og explantion explicitely declare this point.
Re: If n is an integer greater than 6, which of the following mu   [#permalink] 25 Feb 2013, 20:28
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