When positive integer x is divided by positive integer y, the result i

Question

When positive integer x is divided by positive integer y, the result is 59.32. What is the sum of all possible 2-digit remainders for x/y?

A. 560
B. 616
C. 672
D. 728
E. 784

Bunuel · Accepted Answer

VERITAS PREP OFFICIAL SOLUTION:

An important division concept that the GMAT likes to test is that the remainder divided by the divisor becomes the decimal places. (Try this with smaller numbers like 5 divided by 4. The remainder is 1, which when divided by 4 becomes .25. Those are the decimal places in the result 1.25.)

Here you're given the decimal places as .32, which translates to "thirty-two hundredths" or 32/100. Since 32/100 reduces to 8/25, you know that in order to have a two-digit remainder you must have a divisor that's a multiple of 25 and a remainder that's a multiple of 8 and that's a two-digit number. That leaves you with these possibilities:

16/50

24/75

32/100

40/125

etc.

Which should tell you that the possible remainders are all multiples of 8 from 16 to 96. To sum those values, take the number of terms (there are 11 of them, since that's 2 * 8 through 12 * 8) and multiply by the middle value (56, the average of 16 and 96) and the answer is 616.

Divyadisha · Answer

Reminder of x/y= 32/100= 8/25

First two digits multiple of 8 = 16
Last two digit multiple of 8 = 96 (8*12)

Total = Average * Total number of digits in the set

Since its a uniformly arranged set, average = 1st digit+last digit/2= 16+96/2
Total number of digits= 11

Total= 16+96/2*11= 616

B is the answer

chetan2u · Answer

ans B 616...
 remainders = .32=32/100=8/25=16/50 and so on..
 so two digit remainders are 16+24+32+....+96..
 =8(2+3+4....+12)=616

10046844 · Answer

Why would we not count the 8/25? I got 624. Wouldn't it be =8(1+2+3+4....12)?

****Nevermind reread question and see that it is only 2 digit answers that we are using so then B=616

DesiGmat · Answer

Here we go---

Remainder/Divisor = decimal part

Say remainder = r

r/y = 0.32 --> 32/100

so we can have 16+24+32+40+48+56+64+72+80+88+96 = 616

option B is correct

PareshGmat · Answer

\frac{32}{100} = \frac{16}{50} = \frac{24}{75} =  .....

Two digit remainder addition = 16+24+32+40+48+56+64+72+80+88+96 = 8*2 + ................ + 8*12 = 8(2+.......+12) = 77*8 = 616

Answer = B

Ivan91 · Answer

x/y = 96.12
x/y ==> 9612/100 ==> x/y = 96 12/100
From that you can see that the reminder is 12. If you make the reminder 9, that means that you multiply 12 by 3/4
Multiply 100 also by 3/4 and you get 75.

Answer B

gracie · Answer

.32 reduces to 8/25, 16/50, 24/75, 32/100....96/300
the sum of all 2 digit multiples of 8≤96=11*(16+96)/2=616
B

ShashankDave · Answer

After I saw the solution..I got to know what exactly the question was asking for..somehow..I feel that this question should ask it in a better way..if thats not the case, then please help me comprehend such abridged questions.

iliavko · Answer

Hi everyone!

A question here...

I don't really get what the Q is asking, since 8/25, 16/50 etc all represent the same exact number! so, how come we are summing numbers from 8 to 96? for what? They would reduce to the simplest fraction 8/25, so how can those be different numbers (remainders in this case)?

vivophoenix · Answer

i found a better explanation

32/100 reduces down to 8/25

you can not use 8 because that is not a double-digit number, but you want to look for everything that could l reduce down to 8/25 which represent all the other numbers that could be the remainers

the easy way is to multiply by all numbers, 1* 8/ 25= 8/25. 2* 8/25 = 16/50
...... 12* 8/25
= 96/ 300, and you can not use 13 because that is greater than two digits.

so basically you then add up all those values in between

saukrit · Answer

Lowest fraction equivalent of the decimal part is 8/25. Now since we are considering only 2 digit remainders, only the numerator 16 onwards will be used and any fraction with numberator greater than 96 will not be considered as it will become more than 2 digits

=\frac{8*2}{25*2}=\frac{16}{50}

=\frac{8*3}{25*3}=\frac{24}{75} 
..
..
..
 =\frac{8*12}{25*12}=\frac{96}{300}

2 digit remainders are 16,24,32,40,48,56,64,72,80,88,96

Sum of above series=   \frac{n*(n1 + nl)}{2}  where  n=  number of terms =11(in our case),  n1= First term of series and  nl= last term of series

=\frac{11*(16+96)}{2} 
 =\frac{11*112}{2} 
=11*56
  =616

ScottTargetTestPrep · Answer

Since x/y = 59.32 = 59 32/100 = 59 8/25, we see that the possible remainders are of the form 8k where k is a positive integer. Therefore, the first two-digit remainder is 8(2) = 16 and the last two-digit remainder is 8(12) = 96. We can now use the formula sum = average x quantity to find the sum of all possible two-digit remainders. We see that average = (16 + 96)/2 = 56 and quantity = (96 - 16)/8 + 1 = 11. Therefore, the sum is:

56 x 11 = 616

Answer: B

Mansoor50 · Answer

chetan2u Hello...

there is a technique which uses the last digits to find the remainders. Where can i read the nuts and bolts of this? thanks in advance

nkme2007 · Answer

The least reminder is 8 for a y of 25.

And the other reminders being 16 (y=50), 24 (y=75), 32 (y=100), 40 (y=125), 48 (y = 150), 56 (y = 175), 64 (y=200), 72 (y=225), 80 (y=250), 88 (y = 275) and 96 (y=300), are of two digits.

The sum of the above reminders = 616.

OA: B

udaypratapsingh99 · Answer

In the problem above:
Remainder = R
Divisor = 7
Decimal = 0.32 = 32/100 = 8/25
Plugging these values into remainder/divisor = decimal, we get:
R/y = 8/25

The resulting equation implies that the remainder must be a MULTIPLE OF 8.

For any EVENLY SPACED SET:
Count = (biggest - smallest)/(increment) + 1.
Average = (biggest + smallest)/2.
Sum = (count)(average).
The INCREMENT is the difference between successive values.

Here, we must sum the 2-digit multiples of 8 between 16 and 96, inclusive.
Since the integers are multiples of 8, the increment = 8.
Thus:
Count = (96-16)/8 + 1 = 11
Average = (96+16)/2= 56
Sum = (count)(average) = 11*56 = integer with a units digit of 6

The correct answer is B.

GMATWhizTeam · Answer

When x is divided by y, the result is 59.32. The decimal part of the result, when multiplied with the divisor, represents the remainder when x is divided by y.

Therefore, remainder = 0.32 * y =  \frac{32 }{ 100}  * y =  \frac{8 }{25}  * y.

Note that remainder is always a non-negative integer, therefore, y should be a multiple of 25.

If y = 25, remainder = 8. Ignore since we are looking for two-digit remainders.

If y = 50, remainder = 16. This is the first two-digit remainder when x is divided by y.

If y = 75, remainder = 24.

We see that the remainders are consecutive multiples of 8, starting with 16 and going on until 96 (since that’s the last 2-digit multiple of 8).
This is an Arithmetic sequence with a common difference of 8.

16 = 8 * 2; 96 = 8 * 12.

Therefore, number of terms, n = (12 – 2) + 1 = 11.

Sum of terms of an Arithmetic sequence =  \frac{n}{2}  (First term + Last term)

Therefore, sum of all the 2-digit remainders =  \frac{11 }{ 2}  (16 + 96) =  \frac{11 }{ 2}  ( 112) = 11 * 56 = 616.

The correct answer option is B.

aishwaryakakkar · Answer

Concept : X= QY+ R
Dividing both sides by Y
X/Y= Q+R/Y
Since X/Y= 59.32 Therefore Q+ R/Y= 59+ 32/100

Now we know R/Y is roughly equal to 1/3 (32/100 = ~33/100 = 1/3) 
Therefore,  3R=Y 
Since we need only 2 digit numbers the maximum value of Y can be 33

To take sumation of all 2 digit remainders we take the sum from 10 to 33
= n/2  
=24/2 
=616

GMATinsight · Answer

Of 59.32
Part responsible for remainder is .32 and 0.32 = 32/100
So Remainder is 32/100 =8/25 (in its least form)

We need only 2 digit numerators
So 
16/50
24/75
32/100
40/125
48/150
56/175
64/200
72/225
80/250
88/275
96/300
 11 terms

Sum from 16 to 96 = (16+96)*(11/2) = 616

When positive integer x is divided by positive integer y, the result i

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

When positive integer x is divided by positive integer y, the result i

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards