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divisible by 11

by nidhis.1408 » Wed Oct 17, 2012 8:41 am
If n = 14,444, which of the following operations will not yield a number that is divisible by 11?

a. n - 1
b. n + 10
c. n - 100
d. n + 100,000
c. n - 100,000

Is there a quick way to solve this?
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by Whitney Garner » Wed Oct 17, 2012 10:28 am
nidhis.1408 wrote:If n = 14,444, which of the following operations will not yield a number that is divisible by 11?

a. n - 1
b. n + 10
c. n - 100
d. n + 100,000
c. n - 100,000

Is there a quick way to solve this?
Hi nidhis.1408!

I don't know that this is really QUICK, but it is faster than long division! Also, I believe that one of your choices is wrong (E) or else we get 2 answers that meet the criteria (will NOT yield a number that is divisible by 11), so I'm thinking it should be -10,000 NOT -100,000 and I will solve it as such).

So the "trick" to tell divisibility by 11 is to go backwards through the digits of the number alternately subtracting then adding its digits. This can be a bit complicated to explain so let me use a dummy number:

123,456

To test if this number is divisible by 11, I go backwards alternating between subtract and add (always start with subtract)

6-5+4-3+2-1 = 1+1+1 = 3 which is NOT divisible by 11, so neither is this number.

What about 31834:
4-3+8-1+3 = 11 which is divisible by 11 and so is 31,834.

Now we can check the number you have been given:

14444... 4-4+4-4+1 = 1 so this number is NOT divisible by 11 (it is 1 off).

(a) 14443... 3-4+4-4+1 = 0 so this number IS divisible by 11!

(b) 14454... 4-5+4-4+1 = 0 so this number IS divisible by 11! (notice that both A and B made alterations that resulted in a negative 1...3-4 and 4-5 to cancel with that +1 that was making us not divisible by 11

(c) 14544... 4-4+5-4+1 = 2 so this number IS NOT divisible by 11 (notice that this alteration ADDED an additional 1 (because the 5 was in a SUM position) so it didn't cancel with our 1 we were off originally)

(d) 114444... 4-4+4-4+1-1=0 so this number IS divisible by 11!

(e) 4,444... 4-4+4-4=0 so this number IS divisible by 11!

That means that our answer is [spoiler]C[/spoiler]!

Hope this helps!
:)
Whit
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by LalaB » Wed Oct 17, 2012 9:35 pm
answer choice C says n minus 100 ; 14444-100=14344 it is divisible by 11 (check it by summing up digits in odd places,then digits in even places. if the sum of digits in odd places is equal to the sum of digits in even places, then the number is divisible by 11)

1+3+4=4+4 so, 14544 is divisible by 11

answer choice E is not divisible by 11
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by Brent@GMATPrepNow » Thu Oct 18, 2012 7:00 am
nidhis.1408 wrote:If n = 14,444, which of the following operations will not yield a number that is divisible by 11?
a. n - 1
b. n + 10
c. n - 100
d. n + 100,000
c. n - 100,000
Another approach. We can solve this question by first determining the remainder when 14,444 is divided by 11.

Here's one strategy: find a multiple of 11 that's close to (but less than) 14,444
Well, we know 11,000 is a multiple of 11.
Now subtract 11,000 from 14,444 to get 3444

Now find a multiple of 11 that's close to (but less than) 3444
Well, we know 3300 is a multiple of 11.
Now subtract 3300 from 3444 to get 144

Now find a multiple of 11 that's close to (but less than) 144
Well, we know 110 is a multiple of 11.
Now subtract 110 from 144 to get 34

Now find a multiple of 11 that's close to (but less than) 34
Well, we know 33 is a multiple of 11.
Now subtract 33 from 34 to get 1

Great, we now know that, when we divide 14,444 by 11, the remainder is 1.

IMPORTANT: We can now say that 14,444 = 11k + 1 for some integer value of k.
In other words, n = 11k + 1 for some integer value of k.
Notice that we don't need to know the value of k here.

Okay, now let's check the answer choices:

A) n - 1
If n = 11k + 1, then n-1 = 11k
Since 11k is definitely divisible by 11, we can eliminate A.

B) n + 10
If n = 11k + 1, then n+10 = 11k + 11 = 11(k+1)
Since 11(k+1) is definitely divisible by 11, we can eliminate B.

C) n - 100
If n = 11k + 1, then n-100 = 11k - 99 = 11(k-9)
Since 11(k-9) is definitely divisible by 11, we can eliminate C.

D) n + 100,000
If n = 11k + 1, then n+100,000 = 11k + 100,001 = 11(k + 9091)
Since 11(k + 9091) is definitely divisible by 11, we can eliminate D.

This leaves us with answer choice E?
Are we going to check to see whether it's divisible by 11?
No. We don't have the luxury of time to do so.
Take E and move on!

Cheers,
Brent
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by santusri2001 » Thu Oct 18, 2012 7:01 am
Another approach:

Lets assume n-1 is divisible by 11 and check other ans to find which one is not divisible by 11 as following

n+10 = (n-1)+11 =divisible as both n-1 and 11 are divisible
Same way;
n-100= (n-1)-99= divisible
n+100000 = (n-1) + 100001 = divisible
n-100000 = (n-1)- 99999 = not divisible as 99999 is not divisible by 11

So the and is E
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by AIM TO CRACK GMAT » Thu Oct 18, 2012 9:53 am
Brent@GMATPrepNow wrote:
nidhis.1408 wrote:If n = 14,444, which of the following operations will not yield a number that is divisible by 11?
a. n - 1
b. n + 10
c. n - 100
d. n + 100,000
c. n - 100,000
Another approach. We can solve this question by first determining the remainder when 14,444 is divided by 11.

Here's one strategy: find a multiple of 11 that's close to (but less than) 14,444
Well, we know 11,000 is a multiple of 11.
Now subtract 11,000 from 14,444 to get 3444

Now find a multiple of 11 that's close to (but less than) 3444
Well, we know 3300 is a multiple of 11.
Now subtract 3300 from 3444 to get 144

Now find a multiple of 11 that's close to (but less than) 144
Well, we know 110 is a multiple of 11.
Now subtract 110 from 144 to get 34

Now find a multiple of 11 that's close to (but less than) 34
Well, we know 33 is a multiple of 11.
Now subtract 33 from 34 to get 1

Great, we now know that, when we divide 14,444 by 11, the remainder is 1.

IMPORTANT: We can now say that 14,444 = 11k + 1 for some integer value of k.
In other words, n = 11k + 1 for some integer value of k.
Notice that we don't need to know the value of k here.

Okay, now let's check the answer choices:

A) n - 1
If n = 11k + 1, then n-1 = 11k
Since 11k is definitely divisible by 11, we can eliminate A.

B) n + 10
If n = 11k + 1, then n+10 = 11k + 11 = 11(k+1)
Since 11(k+1) is definitely divisible by 11, we can eliminate B.

C) n - 100
If n = 11k + 1, then n-100 = 11k - 99 = 11(k-9)
Since 11(k-9) is definitely divisible by 11, we can eliminate C.

D) n + 100,000
If n = 11k + 1, then n+100,000 = 11k + 100,001 = 11(k + 9091)
Since 11(k + 9091) is definitely divisible by 11, we can eliminate D.

This leaves us with answer choice E?
Are we going to check to see whether it's divisible by 11?
No. We don't have the luxury of time to do so.
Take E and move on!

Cheers,
Brent


Brent its a gud method but dnt u think its 2 long?????????? is der any other quicker method 2 solve d same... i liked the way u elaborated al d steps!!! could u suggest a quicker method?
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by Brent@GMATPrepNow » Thu Oct 18, 2012 10:00 am
AIM TO CRACK GMAT wrote: Brent its a gud method but dnt u think its 2 long?????????? is der any other quicker method 2 solve d same... i liked the way u elaborated al d steps!!! could u suggest a quicker method?
I don't think this approach would take more than 2 minutes. Perhaps it looks longer because I list every little step. But most of the work can be done quickly in your head.

Another option is to memorize the rule for divisibility by 11 (I'm just not very good at memorizing things . . . just ask my wife :-))

Cheers,
Brent
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by LalaB » Thu Oct 18, 2012 10:07 am
I am not a wife,but I already posted the rule for divisibility by 11 :)

the quickest method is to guess and move on (kidding!)

Brent, I liked ur post. This shows again that most of the gmat questions can be solved in multiple ways. that is why we love (do we!? I am a liar hehe) the gmat )))
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by pemdas » Tue Oct 30, 2012 8:38 am
nidhis.1408 wrote:If n = 14,444, which of the following operations will not yield a number that is divisible by 11?

a. n - 1
b. n + 10
c. n - 100
d. n + 100,000
c. n - 100,000

Is there a quick way to solve this?
I've looked through all posts, I guess santusri2001 has posted similar idea
From the first look, it's clear 4,444 all digits of four are repeated, hence the number is divisible by 11. By dealing with 10,000 we can check the choices for divisibility by 11 further. The divisibility by 11 of non-repeated and non-adjacent digit numbers (unlike 3333 or 4444) in itself is checked against the number to contain certain quantity of tens and ones (as units), e.g. 100101 contains 10^5, 10^2 and 1. So by multiplying 11 onto (10^5 + 10^2 +1) we get the whole number equal to 100101.

a. n - 1 <=> 10,000-1 is divisible by 11 because in 9,999 all digits are replicated and adjacent
b. n + 10 <=> 10,000+10 is divisible by 11 because we have 10^4 + 10 quantity of tens and zero ones
c. n - 100 <=> 9,900 is divisible by 11 because we have two digits replicated and adjacent
d. n + 100,000 <=> 110,000 is divisible by 11 because we have two digits replicated and adjacent
e. n - 100,000 <=> -90,000 isn't divisible by 11 because in a new number (90,000) the other digit(s) following (neither adjacent nor in the ten's unit) is not replicated <--- major takeaway from this question.

in e) if it were n - 109,000 we would have divisibility by 11 Or if it were n - 100,090 again the number would be divisible by 11. But not in case of n - 100,009 as we have no replication in the ten's unit.
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