If x and y are both positive integers, what is the remainder when 2*10

Question

If both x and y are positive integers, what is the remainder when  2*10^x + 11y  is divided by 9?

(1) x = 6
(2) y = 8

Bunuel · Accepted Answer

OFFICIAL SOLUTION:

If both x and y are positive integers, what is the remainder when  2*10^x + 11y  is divided by 9?

For an integer to be divisible by 9, the sum of its digits must be divisible by 9. Now, the sum of the digits of 2*10^x (20, 200, 2,000, ...) is always 2, irrespective of x. Thus we need only the value of 11y to determine divisibility of  2*10^x + 11y .

(1) x = 6. Not sufficient.

(2) y = 8. In this case 11y = 88, so the sum of the digits of  2*10^x + 11y  is 2 + (8 + 8) = 18 = {a multiple of 9}, therefore  2*10^x + 11y  IS divisible by 9. Sufficient.

Answer: B.

thakapil007 · Answer

a) 11y/9 would give different remainder for different values of y.  Not sufficient. 
b) 2*10^x /9 always yields 2 as a remainder.  Sufficient .
Hence  B

askhere · Answer

Divisibility rule of 9  
"If the sum of digits of the given number is divisible by 9 then the number itself is divisible by 9."

2*10^x+11y

Stmt 1: x = 6

Whatever the value of x (x is a positive integer) sum of digits of 2*10^x is always 2. (Eg: when 20, 200, 2000 etc)
Depending on the value of y the sum of digits may or may not leave a remainder.
(For eg : if y = 4, 2*10^x+11y = 2000044, sum of digits = 10; not divisible by 9; if y = 8, 2*10^x+11y = 2000088, sum of digits = 18; divisible by 9;
As long as we don't know the value of y we cant say if the expression will leave a remainder or not. 
 Not suff

Stmt 2: y = 8
11y = 88 and sum of digits = 16. Whatever the value of x (x is a positive integer) sum of digits of 2*10^x is always 2.
So sum of digits of 2*10^x+11y is always 18 when y =8. Remainder will be zero.
 
Suff

Answer :  B

Ambarish

Jackal · Answer

Hello all

My attempt:

I like to solve such questions by finding out remainders in the given equation and reducing the remainders successively.

First term
 10^x mod (9) = 1 
Therefore remainder of the first term will always be  2  irrespective of the value of  x

Second term
We need to know the value of  y  to find out the remainder of this term  11 mod(9) = 2  but we need to know  y .

Thus statement 2 is required to answer this question. Answer I will go for is B

peachfuzz · Answer

First of all, a C answer here seems like a trap. There must be a value where the remainder will be unchanged given one of the variables

(1) x = 6
testing values out, the remainder is not dependent on the value of x 
Insufficient

(2) y = 8
No matter the value of x, the remainder is constant when y is given
Sufficient

Answer: B

vishwaprakash · Answer

( 2∗10 ^x)/9 remainder will always be 2. 
Plug in value of x = 0 then remained is 2, X = 2 200/9 remainder is 2.

11y/9 can be broken as (9y +2Y)/9 hence remainder will depend on value of Y only not on X.

Hence Option B.

D3N0 · Answer

Ans: B
we know that 2*10^x, when divided by 9, always gives 2 as reminder. so this question requires us to find the value of y.
Solution:
1) remainder does not depend on the value of x so  
2) now from this we know the value of y so we can find the exact remainder.  
 Statement 2 alone is sufficient to ans the question.

hdwnkr · Answer

This question check focus and presence of mind more than it checks mathematical concept.

Rem (10/9) = 1
Rem (100/9) = 1
Rem (1000/9) = 1

For any positive integer value of x, the remainder Rem ( 2*10^x) is same = 2

In order to have the complete value of the equation, we need y.

Statement 1 is reinstating the fact that is already deduced from the premise. Hence unnecessary and insufficient as it does not give any detail about y

Statement 2 gives a value for y

Hence B

bumpbot · Answer

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.

If x and y are both positive integers, what is the remainder when 2*10

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

If x and y are both positive integers, what is the remainder when 2*10

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards