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# If x, y, and z are positive integers, what is the remainder when 100x

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Re: If x, y, and z are positive integers, what is the remainder when 100x [#permalink]
rpmodi
I think it should be E

you can represent 100X+10Y+Z as a three digit number XYZ

now from condition 1 and 2 , this three digit number is X63 , plug in different values for X : 163/7 --remainder is 2 , 263/7 ----remainder 4

One more comment on this solution : X , Y , Z can be > 10 ,so represtating 100X+10Y+Z as XYZ will not always be right , but by limiting the scope of X, Y, Z values we can simplyfy the solution

Thanks rpmodi, smart explaination!and E is OA
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Re: If x, y, and z are positive integers, what is the remainder when 100x [#permalink]
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sondenso
If x, y, and z are positive integers, what is the remainder when 100x + 10y + z is divided by 7 ?

(1) y = 6
(2) z = 3

Hi,

I set this problem up in a slightly different way.

I noticed that it has 1 equation and 4 unknowns, if you include remainder A.

Stmt 1 = With that logic, you substitute 6, you still have 3 unknowns and one equation. NS - Eliminate A
Stmt 2 = Set up the same equation by substituting 3, you still have 3 unknowns and one equation. NS - Eliminate B

If you set up the equation to include y and z (100x +10 (6) + 3)/7 = a) you have one equation and two unknowns Eliminate C.

Therefore, the only possible solution is E.

Does my logic make sense?
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Re: If x, y, and z are positive integers, what is the remainder when 100x [#permalink]
sondenso
If x, y, and z are positive integers, what is the remainder when 100x + 10y + z is divided by 7 ?

(1) y = 6
(2) z = 3

10y when divided by 7 can leave different remainders.
19*6 = 60 leaves remainder 4

z = 3 leaves remainder 3

but 100x when divided by 7 can leave different remainders. As we don't have concrete value for x, the answer is E
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Re: If x, y, and z are positive integers, what is the remainder when 100x [#permalink]
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100x+10y+z
a)y=6. if we take x,y as same and keep on changing z, we will get different remainders Thus insufficient.
b)z=3. keep x same and z as 6. 10y=10,20,30....
10 remainder = 3
20 remainder = 6
Insufficient.
a&b together) 100x+60+3 = 100x+63. 63 is divisible by 7, thus remainder is based on 100x
x=1, 100 remainder=2
x=2, 200 remainder=4
Hence E
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Re: If x, y, and z are positive integers, what is the remainder when 100x [#permalink]
rpmodi
I think it should be E

you can represent 100X+10Y+Z as a three digit number XYZ

now from condition 1 and 2 , this three digit number is X63 , plug in different values for X : 163/7 --remainder is 2 , 263/7 ----remainder 4

One more comment on this solution : X , Y , Z can be > 10 ,so represtating 100X+10Y+Z as XYZ will not always be right , but by limiting the scope of X, Y, Z values we can simplyfy the solution

Hi - Why can we represent 100X+10Y+Z as a three digit number?
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Re: If x, y, and z are positive integers, what is the remainder when 100x [#permalink]
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mhill5446
rpmodi
I think it should be E

you can represent 100X+10Y+Z as a three digit number XYZ

now from condition 1 and 2 , this three digit number is X63 , plug in different values for X : 163/7 --remainder is 2 , 263/7 ----remainder 4

One more comment on this solution : X , Y , Z can be > 10 ,so represtating 100X+10Y+Z as XYZ will not always be right , but by limiting the scope of X, Y, Z values we can simplyfy the solution

Hi - Why can we represent 100X+10Y+Z as a three digit number?

This is a way of writing an x-digit number.

For example, any two-digit integer can be represented as 10a+b (where a and b are single digit integers), for example 37=3*10+7, 88=8*10+8, etc.
Any three-digit integer can be represented as 100a+10b+c (where a, b and c are single digit integers), for example 371=3*100+7*10+1, ...

Hope it's clear.
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Re: If x, y, and z are positive integers, what is the remainder when 100x [#permalink]
E... since we have no idea of what x is

Sent from my GT-I9060I using GMAT Club Forum mobile app
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Re: If x, y, and z are positive integers, what is the remainder when 100x [#permalink]
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sondenso
If x, y, and z are positive integers, what is the remainder when 100x + 10y + z is divided by 7 ?

(1) y = 6
(2) z = 3

We need to determine the remainder of (100x + 10y + z)/7. If we can determine the remainder of 100x/7, 10y/7, and z/7, then we can determine the remainder of (100x + 10y + z)/7.

Statement One Alone:

y = 6

Using the information in statement one, we have:

(100x + 60 + z)/7

Although we know the remainder of 60/7 is 4 (note: 60/7 = 8 + 4/7), we still cannot determine the remainder of 100x/7 or z/7. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

z = 3

Using the information in statement two, we have:

(100x + 10y + 3)/7

Although we know the remainder of 3/7 is 3, we still cannot determine the remainder of 10y/7 or 100x/7.

Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using the information from statements one and two, we have:

(100x + 60 + 3)/7 = (100x + 63)/7

Although we know the remainder of 63/7 is 0, we still cannot determine the remainder of 100x/7. Different values of x might yield different remainders. For example, if x = 1, then the remainder of 100/7 is 2, since 100/7 = 14 + 2/7. However, if x = 2, then the remainder of 200/7 is 4, since 200/7 = 28 + 4/7.

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Re: If x, y, and z are positive integers, what is the remainder when 100x [#permalink]
I had guessed E because we are given only two of three variables with the two statements. I didn't consider testing values and/or reasoning it out - moreso guessed it straight away because we are still missing one variable within the equation (similar to hsingh2008's post above) -- is this a valid approach or no?
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If x, y, and z are positive integers, what is the remainder when 100x [#permalink]
The number XYZ is devided by 7 if 2*Z-XY can be devided by 7. In this case we have got only Z and Y but not X. So the answer is E. Another way to silve is to take 163 and 263 and compare remainders - they are different.
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If x, y, and z are positive integers, what is the remainder when 100x [#permalink]
Bunuel
mhill5446
rpmodi
I think it should be E

you can represent 100X+10Y+Z as a three digit number XYZ

now from condition 1 and 2 , this three digit number is X63 , plug in different values for X : 163/7 --remainder is 2 , 263/7 ----remainder 4

One more comment on this solution : X , Y , Z can be > 10 ,so represtating 100X+10Y+Z as XYZ will not always be right , but by limiting the scope of X, Y, Z values we can simplyfy the solution

Hi - Why can we represent 100X+10Y+Z as a three digit number?

This is a way of writing an x-digit number.

For example, any two-digit integer can be represented as 10a+b (where a and b are single digit integers), for example 37=3*10+7, 88=8*10+8, etc.
Any three-digit integer can be represented as 100a+10b+c (where a, b and c are single digit integers), for example 371=3*100+7*10+1, ...

Hope it's clear.

Bunuel,

Hi i wanted to understand how did we come to know that its a three digit number ?

But we are told that that X,Y,Z are positive integers. Had we be given that they are digits we can express 100X+10Y+Z as three digit positive integer.

Here is what i mean if X= 2 Y=6 Z=3 then the number would be 263

But if X = 20 Y=6 and Z 3 then the number would be 2063.

So with certainty how can we say that its a three digit positive number, while we only know that X,Y,Z are positive integers.

But i had a different approach for this question

n= 100X+10Y+Z
n= 98X+2X+7Y+3Y+Z
$$\frac{n}{7}$$= $$\frac{98X+2X+7Y+3Y+Z}{7}$$

so remainder will be $$\frac{2X+3Y+Z}{7}$$

So to answer the question we need to know all the three variable X,Y,Z

Originally posted by Probus on 15 Mar 2019, 08:09.
Last edited by Probus on 04 Apr 2019, 08:38, edited 2 times in total.
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Re: If x, y, and z are positive integers, what is the remainder when 100x [#permalink]
Hi Probus,

You're correct - there is nothing in the prompt that states that the final number must be a 3-digit number. By thinking in those terms however, the work becomes a lot easier - and you can quickly prove that Fact 1 and Fact 2 are each INSUFFICIENT (as well as prove that combining both Facts still leads to an INSUFFICIENT answer).

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Re: If x, y, and z are positive integers, what is the remainder when 100x [#permalink]
hsingh2088
sondenso
If x, y, and z are positive integers, what is the remainder when 100x + 10y + z is divided by 7 ?

(1) y = 6
(2) z = 3

Hi,

I set this problem up in a slightly different way.

I noticed that it has 1 equation and 4 unknowns, if you include remainder A.

Stmt 1 = With that logic, you substitute 6, you still have 3 unknowns and one equation. NS - Eliminate A
Stmt 2 = Set up the same equation by substituting 3, you still have 3 unknowns and one equation. NS - Eliminate B

If you set up the equation to include y and z (100x +10 (6) + 3)/7 = a) you have one equation and two unknowns Eliminate C.

Therefore, the only possible solution is E.

Does my logic make sense?

I would love to be able to validate this approach too, as it seems like the fastest solution to me? Thanks
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Re: If x, y, and z are positive integers, what is the remainder when 100x [#permalink]
Info from the stem:
x, y, and z > 0 and integers,

Need to find:
Remainder when 100x + 10y + z is divided by 7.
Rewrite the expression as 98X + 7Y + (2X + 3Y + Z). Now 98X and 7Y are always divisible by 7, so we need to find the remainder when 2X + 3Y + Z is divided by 7, for which we need to know the value of all three variables.

S1: Y = 6. Insufficient
S2: Z = 3. Insufficient

S1 & S2 - The value of X is unknown. Insufficient.

The correct answer is Option E.
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Re: If x, y, and z are positive integers, what is the remainder when 100x [#permalink]
Here's another way to look at the question. There is no pattern or trend we can find, when dividing a number or expression by 7. This is in contrast to how numbers or expressions are divisible by numbers: 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12 (check for divisibility rules).

Yet to be a 100% sure, we can check if Statement 1 is SUFFICIENT or not, and then if Statement 2 is SUFFICIENT or not. In either case, you can plug in multiple values for x or z, or x or y respectively in order to check for the expression to be divisible by 7.

Hence, E is the best choice, IMO.

Let me know if this explanation helps!
Re: If x, y, and z are positive integers, what is the remainder when 100x [#permalink]
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