X and Y are 2-digit positive no. and Z is a 3-digit positive no., such

Question

X  and  Y  are two-digit positive number and  Z  is a three-digit positive number, such that  Z=X+Y . Is the remainder when  X  is divided by  11  less than the remainder when  Y  is divided by  11 ?

(1) All the digits of  Z  are the same, and the remainder when  X  is divided by  70  is two more than the fifth power of a prime number

(2) All the digits of  Y  are the same

freedom128 · Answer

Q. Is the remainder when X is divided by 11 less than the remainder when Y is divided by 11?

(1) All the digits of Z are the same, and the remainder when X is divided by 70 is two more than the fifth power of a prime number 
Z=X+Y, where Z is a three-digit positive number. This means Z<200 (consider adding the maximum of 99 to 99: It's always less than 200).
Statement (1) simply means that  Z=111  and  X=2^5+2=34  (anything beyond  2^5  won't result in two-digit X). We can then determine  Y  from  Z=X+Y .
SUFFICIENT

(2) All the digits of Y are the same 
Statement (1) means that  Y  could be  11, 22,...,99 . As a result, the remainder when Y is divided by 11 is always 0 (zero).
For arbitrary value of two-digit integer X, the remainder when X is divided by 11  \geq{0} . 
Therefore, the remainder when X is divided by 11 is NEVER less than the remainder when Y is divided by 11.
SUFFICIENT

Answer is (D)

unraveled · Answer

You changed the question statement wording ..

. it was like this X and Y are two-digit positive number and Z is a three-digit positive number, such that Z=X+Y. Is the remainder when X is divided by 11 less than the remainder when Y is divided by 11? (1) All the digits of Z are the same, and all the digits of Y are the same (2) The remainder when X is divided by 70 is the fifth power of a prime number I had so many doubts with those statements. I have to do it again

freedom128 · Answer

Haha, sorry about that. The answer is actually the same with the current answer. Find similar topical questions for practice here: 1) https://gmatclub.com/forum/if-a-and-b-a ... l#p1375424 2) https://gmatclub.com/forum/if-a-and-b-a ... l#p2011550 3) https://gmatclub.com/forum/if-a-and-b-a ... l#p2351764 4) https://gmatclub.com/forum/if-n-is-a-no ... l#p2349427

chetan2u · Answer

chondro48

X  and  Y  are two-digit positive number and  Z  is a three-digit positive number, such that  Z=X+Y . Is the remainder when  X  is divided by  11  less than the remainder when  Y  is divided by  11 ?

You can say for sure that X and Y are LESS than 100 and atleast one is greater than 50, since their sum is 3-digit number..
Also, if both X and Y were the largest possible values their SUM would be 99+99=198, so  Z\leq{198}

(1) All the digits of  Z  are the same, and the remainder when  X  is divided by  70  is two more than the fifth power of a prime number
We know Z is 3-digit and  Z\leq{198} .  So, if three digits are same, only possibility for Z is 111, as hundreds digit is surely 1.

WHAT about X?  
Remainder when a number is divided by 70 has to be LESS than 70. So check for 5th power of a prime number that is less than 67 ( 69-2), as the remainder is  x^5+2 .  Only 2 fits in  as  2^5=32 , while  3^5>70 . 
 Thus remainder is 70z+32+2...  
 Possible values are  
when z =0, so 34 ...X=34 and Y=111-34=77..X will leave a bigger remainder
when z=1, so 104....X=104, and Y=111-104=7... NOT possible as Y becomes a single digit number. 
 Only one answer X=34 and Y=77 
Sufficient

(2) All the digits of  Y  are the same
If Y has same digits, and 2-digit number,  it will always be divisible by 11. 
Hence, the answer for - "Is the remainder when  X  is divided by  11  less than the remainder when  Y  is divided by  11 " will always be NO, as  the remainder will always  be either EQUAL, if X is divisible by 11  OR  MORE  . 
Suff

D

rajshreeasati · Answer

Nice question!
 We know that x and y are both 2 digit nos and z is 3 digit, so minimum z is 100 and minimum x and y is 10. We know Z = x + y.
Now, X = 11m + r and Y =11n + R.
Need to find, r<R. Y/N.

Stt1. Z can be 111,222,333 etc. But if we take Z as 222 or 333 etc, then x or y becomes 3 digit, which we dont want, so definitely z is 111. Now remainder when x iw divided by 70 is 2 + p^5. And let x = 34 and y =77,then Z =111. When x is 34, 34/70 is 34, which is 2+32, ie.x is definitely 34. This remainder when 34/11 is 1 and 77/11 is 0 thus stt1 is sufficient.

Stt2) all digits of y are same. So y can be 11, 22, 33, 44 etc. And whatever its value is, the remainder when y divided by 11 us always 0 and x is lets say 90 thus the remainder is always more than 0. Thus stt 2 is also sufficient

Posted from my mobile device

rajshreeasati · Answer

Nice question!
 We know that x and y are both 2 digit nos and z is 3 digit, so minimum z is 100 and minimum x and y is 10. We know Z = x + y.
Now, X = 11m + r and Y =11n + R.
Need to find, r<R. Y/N.

Stt1. Z can be 111,222,333 etc. But if we take Z as 222 or 333 etc, then x or y becomes 3 digit, which we dont want, so definitely z is 111. Now remainder when x iw divided by 70 is 2 + p^5. And let x = 34 and y =77,then Z =111. When x is 34, 34/70 is 34, which is 2+32, ie.x is definitely 34. This remainder when 34/11 is 1 and 77/11 is 0 thus stt1 is sufficient.

Stt2) all digits of y are same. So y can be 11, 22, 33, 44 etc. And whatever its value is, the remainder when y divided by 11 us always 0 and x is lets say 90 thus the remainder is always more than 0. Thus stt 2 is also sufficient

Posted from my mobile device

mSKR · Answer

Happy New year

It is easy to identify the values from A 
Z= 111
X= can be 34 or 104
As Y is 2 digit, so it means X= 34 ; Y = 77
Hence A is sufficient

For B , it is not sufficient to idendity the values BUT what's the question?
  Is the remainder when X is divided by 11 less than the remainder when Y is divided by 11? 
Given: Less than ; not even less than or equal to 
It means remainder can be 0 or greater than 0
in no case , it should be less than 0,.

hence B sufficient

Final answer: D

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X and Y are 2-digit positive no. and Z is a 3-digit positive no., such

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