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Math Expert V
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If x and yare positive integers , what is the remainder when  [#permalink]

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The Official Guide For GMAT® Quantitative Review, 2ND Edition

If x and y are positive integers, what is the remainder when $$10^x + y$$ is divided by 3 ?

(1) x = 5
(2) y = 2

Data Sufficiency
Question: 16
Category: Arithmetic Properties of numbers
Page: 154
Difficulty: 600

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Math Expert V
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Re: If x and yare positive integers , what is the remainder when  [#permalink]

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SOLUTION

If x and y are positive integers, what is the remainder when 10^x +y is divided by 3?

Since, the sum of the digits of 10^x is always 1 then the remainders when 10^x+y is divided by 3 is only dependant on the value of the number added to 10^x, so on y. If y is a multiple of 3 then 10^x+y will yield the remainder of 1 (since the sum of the digits of 10^x+y will be one more than a multiple of 3), if y is one more than a multiple of 3 then 10^x+y will yield the remainder of 2 and finally if y is two more than a multiple of 3 then 10^x+y will yield the remainder of 0,

(1) x = 5. Not sufficient.
(2) y = 2. Sufficient.

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Joined: 27 May 2012
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Re: If x and yare positive integers , what is the remainder when  [#permalink]

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Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

If x and y are positive integers, what is the remainder when 10^x + y is divided by 3 ?

(1) x = 5
(2) y = 2

We can see that 10^x + y will always be of the form 10+y , or 100 +y or 1000+y etc depending on the power of x
so 1 is always going to be carried forward from the 10^x side to y

e.g.

if x= 1 and y = 1 then 10^x+y will give 10+1 =11, divided by 3 remainder 2
if x= 2 and y = 1 then we have 100+1 =101,divided by 3 remainder 2
if x=2 and y =2 then we have 100+2 =102,divided by 3 remainder 0
etc

hence the remainder really depends on the value of y

1) if y = 1 remainder is 2, if y is 2 remainder is 0, since no info about y hence insuff.

2)Y =2 , just what we were looking for , no matter what ever is the value of x remainder will always be 0.
as 10^x+y will always be of the form:

12
102
1002
10002
etc remainder in all cases is 0 hence B is suff
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- Stne

Originally posted by stne on 04 Jan 2014, 02:05.
Last edited by stne on 10 May 2015, 17:43, edited 1 time in total.
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Re: If x and yare positive integers , what is the remainder when  [#permalink]

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Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

If x and y are positive integers, what is the remainder when 10^x + y is divided by 3 ?

(1) x = 5
(2) y = 2

Before diving into the statements, lets try to simplify the expression. (10^x + y) rem by 3 = (1^x + y) rem by 3 = y rem by 3. The rule of simplification is applicable only in case of remainders.

So clearly Statement is Sufficient. hence Option B)
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Re: If x and yare positive integers , what is the remainder when  [#permalink]

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Great, but it should be (10^x) + y. Otherwise it is as if x+y is the power of 10.
Math Expert V
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Re: If x and yare positive integers , what is the remainder when  [#permalink]

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pacifist85 wrote:
Great, but it should be (10^x) + y. Otherwise it is as if x+y is the power of 10.

It should be as it is. If it were $$10^{x+y}$$, then it would be written as 10^(x+y).
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Re: If x and yare positive integers , what is the remainder when  [#permalink]

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stne wrote:
Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

If x and y are positive integers, what is the remainder when 10^x + y is divided by 3 ?

(1) x = 5
(2) y = 2

We can see that 10^x + y will always be of the form 10+y , or 100 +y or 1000+y etc depending on the power of x
so 1 is always going to be carried forward from the 10^x side to y

e.g.

if x= 1 and y = 1 then 10^x+y will give 10+1 =11, divided by 3 remainder 2
if x= 2 and y = 1 then we have 100+1 =101,divided by 3 remainder 2
if x=2 and y =2 then we have 100+2 =102,divided by 3 remainder 0
etc

hence the remainder really depends on the value of y

1) if y = 1 remainder is 2, if y is 2 remainder is 0, since no info about y hence insuff.

2)X=2 , just what we were looking for , no matter what ever is the value of x remainder will always be 0.
as 10^x+y will always be of the form:

12
102
1002
10002
etc remainder in all cases is 0 hence B is suff

Not X=2, Y=2. Please correct a typo=))
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Re: If x and yare positive integers , what is the remainder when  [#permalink]

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Bunuel wrote:
SOLUTION

If x and y are positive integers, what is the remainder when 10^x +y is divided by 3?

Since, the sum of the digits of 10^x is always 1 then the remainders when 10^x+y is divided by 3 is only dependant on the value of the number added to 10^x, so on y. If y is a multiple of 3 then 10^x+y will yield the remainder of 1 (since the sum of the digits of 10^x+y will be one more than a multiple of 3), if y is one more than a multiple of 3 then 10^x+y will yield the remainder of 2 and finally if y is two more than a multiple of 3 then 10^x+y will yield the remainder of 0,

(1) x = 5. Not sufficient.
(2) y = 2. Sufficient.

Bunuel

〖10〗^(x+y)/3 => 〖10〗^x/3 X 〖10〗^y/3 => 10/3 X 10/3 => 1 X 1 => 1 (Remainder) --- (1)

〖10〗^x/3+y/3 => 〖10〗^ /3+y/3 => 1 + (y/3) , given y=2 => 1 + 2 = 3 ( Remainder) ----- (2)

Please tell which one above the question is referring to and also please if my approach is wrong. Considering (2) , answer shall be B.
Math Expert V
Joined: 02 Sep 2009
Posts: 58312
Re: If x and yare positive integers , what is the remainder when  [#permalink]

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merajul wrote:
Bunuel wrote:
SOLUTION

If x and y are positive integers, what is the remainder when 10^x +y is divided by 3?

Since, the sum of the digits of 10^x is always 1 then the remainders when 10^x+y is divided by 3 is only dependant on the value of the number added to 10^x, so on y. If y is a multiple of 3 then 10^x+y will yield the remainder of 1 (since the sum of the digits of 10^x+y will be one more than a multiple of 3), if y is one more than a multiple of 3 then 10^x+y will yield the remainder of 2 and finally if y is two more than a multiple of 3 then 10^x+y will yield the remainder of 0,

(1) x = 5. Not sufficient.
(2) y = 2. Sufficient.

Bunuel

〖10〗^(x+y)/3 => 〖10〗^x/3 X 〖10〗^y/3 => 10/3 X 10/3 => 1 X 1 => 1 (Remainder) --- (1)

〖10〗^x/3+y/3 => 〖10〗^ /3+y/3 => 1 + (y/3) , given y=2 => 1 + 2 = 3 ( Remainder) ----- (2)

Please tell which one above the question is referring to and also please if my approach is wrong. Considering (2) , answer shall be B.

It's $$10^x + y$$ not $$10^{x + y}$$.
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_________________ Re: If x and yare positive integers , what is the remainder when   [#permalink] 05 Dec 2018, 20:44
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