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

Question

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

EMPOWERgmatRichC · Accepted Answer

Hi All,

This post is from years ago, but the prompt still serves as an example of how TESTing VALUES can help you to prove the correct answer in many DS questions.

We're told that X, Y and Z are POSITIVE INTEGERS. We're asked for the REMAINDER when 100X + 10Y + Z is divided by 7.

Fact 1: Y = 6

Since we don't know the values of X or Z, let's TEST VALUES.

Technically, we can use ANY positive integers for X and Z, but I'm going to keep things simple...
X = 1, Z = 1

(100 + 60 + 1)/7
161/7 = 23r0 so the answer is 0

X = 1, Z = 3
(100 + 60 + 3)/7
163/7 = 23r2 so the answer is 2
Fact 1 is INSUFFICIENT

Fact 2: Z = 3

Again, we can use ANY positive integers for X and Y, but let's keep things simple...

We can use the second example from Fact 1 here...
X = 1, Y = 6, Z = 3
163/7 = 23r2 so the answer is 2

X = 1, Y = 1, Z = 3
(100 + 10 + 3)/7
113/7 = 16r1 so the answer is 1
Fact 2 is INSUFFICIENT

Combined, we know
Y = 6
Z = 3
X = ANY POSITIVE INTEGER

If X = 1 (from our prior work)
163/7 = 23r2 so the answer is 2

If X = 2
(200 + 60 + 3)/7
263/7 = 37r4 so the answer is 4
Combined, INSUFFICIENT

Final Answer:  E

GMAT assassins aren't born, they're made,
Rich

rpmodi · Answer

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 ;-)

sondenso · Answer

Thanks rpmodi, smart explaination!and E is OA

hsingh2088 · Answer

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?

shank001 · Answer

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

anairamitch1804 · Answer

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

mhill5446 · Answer

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

Bunuel · Answer

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.

Sumo23 · Answer

E... since we have no idea of what x is

Sent from my GT-I9060I using  GMAT Club Forum mobile app

ScottTargetTestPrep · Answer

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.

Answer: E

infinitemac · Answer

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?

Hero8888 · Answer

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.

Probus · Answer

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

EMPOWERgmatRichC · Answer

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).

GMAT assassins aren't born, they're made,
Rich

TargetMBA007 · Answer

I would love to be able to validate this approach too, as it seems like the fastest solution to me? Thanks

MariaAntony · Answer

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.

m7runner · Answer

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!

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

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

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