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Integer Properties - DS

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by theadityajamwal » Wed Jul 13, 2011 1:45 am
Q) remainder of 8*10^k + j when divided by 9.
Rephrasing the Q gives me
[( 9 - 1 ) * 10^k + j] / 9
( 9 * 10^k - 10^k + j )/9
Now 9 * 10^k will always be divisible by 9. So we can omit that and are left with
( j - 10^k )/9
[j - (9+1)^k ]/9
Now for any equation (a+b)^n, every term will be divisible by a except b^n.
Eg (a + b)^3 = a^3 + 3 a^2 * b + 3ab^2 + b^3. So every term except b^3 is divisible by a.

Hence our equation modifies to ( j - 1^k )/9
( j - 1 )/9.

Hence all we want is the value of j.

Answer will be B.

This method will be helpful in case we have a question like
what will be the remainder when 12 * 14^k + j is divided by 13.
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by saurabh2525_gupta » Wed Jul 13, 2011 8:14 pm
8*10^k + j can be written as (9-1)*(9+1)^k + j. The polynomial function (9+1)^k will have certain number of terms depending on the value of k. When we expand (9+1)^k, we see that all the terms have a factor of 9 except the last one. For example (9+1)^2 = 9^2 + 2*9*1 + 1^2.

The remainder of (9+1)^k will always be 1 irrespective of the value of k. The remainder from (9-1) will be 8. When we multiply both the individual remainders we get 8 as the remainder from the expression (9-1)*(9+1)^k.

The value 8 is not dependent on the value of k. What will determine the remainder from the complete expression is the value of j. When j =1, the remainder is always 0.

=> Answer is B.

Best Regards,
John
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by emf_jay » Fri Jul 15, 2011 9:45 am
thanQ
ready for 750 score .... !!!
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by shingik » Fri Jul 15, 2011 4:52 pm
(1) is insufficient because j could be any positive integer. We do not need to know what k is becasue 10 raised to any power results in a number that begins with 1 and is followed by k zeroes
(2) is sufficient because 8 * 1 followed by k zeroes is 8 followed by k zeroes. If you add 1 to it you get a number begining with 8 followed by k-1 zeroes and ending with 1. Remembering the general divisible by nine rule. i.e. a number is divisible by nine if the sum of its digits is nine. It is easy to see that 8 and 1 add up to nine so the resultant number is divisble by nine. So B should be the correct choice.
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by shingik » Fri Jul 15, 2011 4:58 pm
saurabh2525_gupta wrote:8*10^k + j can be written as (9-1)*(9+1)^k + j. The polynomial function (9+1)^k will have certain number of terms depending on the value of k. When we expand (9+1)^k, we see that all the terms have a factor of 9 except the last one. For example (9+1)^2 = 9^2 + 2*9*1 + 1^2.

The remainder of (9+1)^k will always be 1 irrespective of the value of k. The remainder from (9-1) will be 8. When we multiply both the individual remainders we get 8 as the remainder from the expression (9-1)*(9+1)^k.

The value 8 is not dependent on the value of k. What will determine the remainder from the complete expression is the value of j. When j =1, the remainder is always 0.

=> Answer is B.

Best Regards,
John
Correct but so complicated. this is just testing if you know that if the sum of a number's digits add up to nine then the number is divisible by nine i.e. 8 000000...1 ----- 8+1=9 so B is sufficient
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by shingik » Fri Jul 15, 2011 5:14 pm
anirudhbhalotia wrote:Fantastic questions..totally squeezed my tiny brains by trying to apply the concepts! But....

...why the options are not mentioned....A, B, C, D, E?
You MUST know the options before you even attempt data sufficiency questions. That is the golden rule. You should know how to eliminate options without really thinking about it. That almost goes without saying. Your score will improve dramatically once you master the structure of DS questions. Your mastery of the structure should be second nature before you even begin working problems. I promise your score will jump dramatically if thus far you have been answering DS questions blindly. Good Luck. I am in the struggle with you and we will make it!!!!!
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by astvansh » Thu Jul 28, 2011 6:03 am
'b'

For a # to be divisible by 9, the sum of digits should be divisible by 9. Now, irrespective of the value of k, the sum of digits of (8*10^k) will be 8. If j is such that 8+j is divisible by 9, the resulting # will be divisible by 9.
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by saketk » Fri Aug 19, 2011 11:08 am
Pretty simple -- option B should suffice.
Divisibility Rule of 9. The sum should be equal to 9 and this is what we get every time if J=1 . :)
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by Deependra1 » Sat Aug 27, 2011 10:19 am
ANSWER:B

Please correct if its incorrect
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by JS_2 » Mon Aug 29, 2011 9:27 pm
The Ans is B as the value of the remainder changes according to the value of j.
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by jsnipes » Tue Sep 13, 2011 9:39 am
is the question 8*10^(k+j) or (8*10^k)+j divided by 9?

i think i was interpreting the question as the former in which case it doesnt matter what the sum of k and j is since the remainder will always be 8. i see how the answer makes sense though if the questions is the latter (in which case, either i am an idiot or i think the question needs to be written clearer imo). i have been sitting here trying to rack my brain to figure out how 8*10^(x+1) has any different remainder than 8*10^x since (8x10^x)/9 will always have a remainder of 8.

am i the only one that misread this question?
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by arunnarang » Sat Oct 08, 2011 6:57 am
ans is 8
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by parul9 » Sat Oct 08, 2011 7:53 am
B!
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by Brent@GMATPrepNow » Wed Oct 12, 2011 7:17 am
jsnipes wrote:is the question 8*10^(k+j) or (8*10^k)+j divided by 9?
am i the only one that misread this question?
Good point, jsnipes - my bad.
One of my biggest beefs are ambiguous questions, and I neglected to add brackets to avoid ambiguity.

I have since added brackets to the original question so that it reads: 8 *(10^k) + j

Cheers and thanks,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by MrCleantek » Wed Oct 12, 2011 9:50 am
Common Brent lets get some more questions!!
Do what your heart says......
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