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# If x and yare positive integers , what is the remainder when

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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

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 y are positive integers, what is the remainder when [#permalink]
The question is not really understandable. Please post it either in the "formula form" or like this (10^x)+y.
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Re: If x and y are positive integers, what is the remainder when [#permalink]
unceldolan wrote:
The question is not really understandable. Please post it either in the "formula form" or like this (10^x)+y.

10^x +y means $$10^x +y$$, so no ambiguity there. If it were 10^(x +y) it would be written that way. Still edited the original post to avoid further confusions.
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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.
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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 y are positive integers, what is the remainder when [#permalink]
Bunuel wrote:
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

I'm sorry if this is something obvious i'm missing but I'm not quite following what you mean by that; sum of digits 10^x is always 1 as in...10 and x?
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Re: If x and y are positive integers, what is the remainder when [#permalink]
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redfield wrote:
Bunuel wrote:
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

I'm sorry if this is something obvious i'm missing but I'm not quite following what you mean by that; sum of digits 10^x is always 1 as in...10 and x?

If x = 1, then 10^1 = 10 --> the sum of the digits = 1 + 0 = 1;
If x = 2, then 10^3 = 100 --> the sum of the digits = 1 + 0 + 0 = 1;
If x = 3, then 10^3 = 1000 --> the sum of the digits = 1 + 0 + 0 + 0= 1;
...

Hope it's clear.
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Re: If x and y are positive integers, what is the remainder when [#permalink]
Bunuel wrote:
redfield wrote:
Bunuel wrote:
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

I'm sorry if this is something obvious i'm missing but I'm not quite following what you mean by that; sum of digits 10^x is always 1 as in...10 and x?

If x = 1, then 10^1 = 10 --> the sum of the digits = 1 + 0 = 1;
If x = 2, then 10^3 = 100 --> the sum of the digits = 1 + 0 + 0 = 1;
If x = 3, then 10^3 = 1000 --> the sum of the digits = 1 + 0 + 0 + 0= 1;
...

Hope it's clear.

Sorry I'm still struggling with this; what are you equating the 1 and the 0's to, the 1 being the 10 and the 0's being the power it is risen to?
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Re: If x and y are positive integers, what is the remainder when [#permalink]
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redfield wrote:
Bunuel wrote:
redfield wrote:
I'm sorry if this is something obvious i'm missing but I'm not quite following what you mean by that; sum of digits 10^x is always 1 as in...10 and x?

If x = 1, then 10^1 = 10 --> the sum of the digits = 1 + 0 = 1;
If x = 2, then 10^3 = 100 --> the sum of the digits = 1 + 0 + 0 = 1;
If x = 3, then 10^3 = 1000 --> the sum of the digits = 1 + 0 + 0 + 0= 1;
...

Hope it's clear.

Sorry I'm still struggling with this; what are you equating the 1 and the 0's to, the 1 being the 10 and the 0's being the power it is risen to?

We are talking about the sum of the digits of a number. For example, the sum of the digits of 17 is 8 because 1 + 7 = 8. Or, the sum of the digits of 2^5 is 5 because 2^5 = 32 and 3 + 2 = 5.
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Re: If x and y are positive integers, what is the remainder when [#permalink]
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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

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

In the original condition, there are 2 variables(x,y), which should match with the number of equations. So, you need 2 equations. For 1) 1 equation, for 2) 1 equation, which is likely to make C the answer. In 1) & 2), it becomes 10^5+2=100,002. Also, the remainder after divided by 3 is same as the remainder after the sum of all digits divided by 3. That is, 1+0+0+0+0+2=3 can be divided by 3 and therefore it is yes, which is sufficient. So the answer is C. This is an integer question which is one of the key questions. When applying 4(A) of the mistake type,
1) x=5 but y=2- > yes, y=3 -> no, which is not sufficient.
2) y=2 -> 10^x+2=12,102,1002,10002......... When it comes to number like these, the sum of all digits are 3 and also can be divided by 3 and it is yes, which is sufficient. Therefore, both B and C can be the answer. When both B and C can be the answer, the answer is B.
This type of question is given in Math Revolution lectures and you need to get this type of question right to reach the score range 50-51.

-> For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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Re: if integers x,y > 0, what is the remainder when (10^x + y)/3? Logic? [#permalink]
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nilem94 wrote:
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

I understood the explanation, and indeed B was my first choice, too.
However, I'm struggling with its logic. Because the questions ask for a value, so I thought it depends on the power of 10th? I see the similar pattern it follows (33,334.. etc), yet different powers would yield in different results, therefore I won't be able to find out the exact value.

i.e. (10^1 + 2)/ 3 = 4 or (10^2 + 2 /3) = 34)

Could you explain, why in I say B is sufficient, even though there are many values? (Find the question in the 2nd Quant Review)

Hi,

INFO:-

since the div rule by 3 states that the sum of digits should be div by 3 and the remainder of any integer is same as remainder of the sum of integer..
lets see what is 10^x + y...
here irrespective of value of x, the sum of integers will be 1, as it will be 1,10,100,1000...
so we require to know y to find the remainder..

lets see the choices..
1) X=5
insuff

(2) y=2
now we know sum = 1+2..
remainder is 0..
Suff
B
Hope it helped
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Re: If x and yare positive integers , what is the remainder when [#permalink]
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dzodzo85 wrote:
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

Solution:

Question Stem Analysis:

We need to determine the remainder when 10^x + y is divided by 3, given that x and y are positive integers. Notice that 10^x is the number 1 followed by x zeros. Since the sum of these digits is 1, the remainder when 10^x is divided by 3 will be 1. Therefore, the answer to the question depends on the value of y. For example, if y = 7, then the remainder will be 2 since the sum of the digits of 10^x + y will be 1 + 7 = 8, and 8 divided by 3 is 2, with a remainder of 2.

Statement One Alone:

Since we don’t know anything about y, statement one is not sufficient.

Statement Two Alone:

Since y = 2, the sum of the digits of 10^x + y will be 1 + 2 = 3, which has a remainder of 0 when divided by 3. Statement two is sufficient.

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