Problem Solving

Most Manhattan GMAT students are trying to break the 700 barrier. As a result, we've developed our own math problems written at the 700+ level; these are the types of questions you'll WANT to see, when you are working at that level. Try to solve this 700+ level problem (I'll post the solution next Monday).

Question:
When the integer x is divided by the integer y, the remainder is 60. Which of the following is a possible value of the quotient x/y?

I. 15.15
II. 18.16
III. 17.17

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

anyone have thoughts on this one? i'm really bad at these kinds of problems...

My best shot:-

Key to the problem is that X & Y are both integers.
So the product of (portion after decimal & remainder) has to be an integer.

I. 15.15 (0.15 X 60= 9 integer..)
II. 18.16 (0.16 X 60= 9.6 Not integer..)
III. 17.17 0.17 X 60= 10.2 Not integer..)

Only I is surely possible.
So my answer is
A) I only

Let me know if this is right approach.

thanks gmat_score! i guess we'll find out the answer tomorrow when kevin makes his post...

I think the right answer is D.
Here is the solution as per my understanding -

x/y = m(60/y) [read as m whole 60 divided by y] where x, m and y (>60) are integers. And 60/y has to be equal to the fractional part. Lets try one by one.

I.
60/y = 0.15
implies, y = 400 (m is the integral part = 15) Correct.

II.
60/y = 0.16
implies, y =375 ( m=18 ) Correct.

III.
60/y =0.17
gives us y non-integer. Hence incorrect.

This makes us choose I and II only making answer choice D to the correct one. Lets keep our fingers crossed and wait ...

Answer:If the integer x divided by y has a remainder of 60, then x can be expressed as:
x = ky + 60, where k is an integer (i.e. y goes into x k times with a remainder of 60)

We could also write an expression for the quotient x/y:

x=k + 60
y y




Notice that k is still the number of times that y goes into x evenly.
60
y
is the decimal portion of the quotient, i.e. the remainder over the divisor y.

The first step to solving this problem is realizing that k, the number of times that y goes into x evenly, can be anything for this question since we are only given a value for the remainder. The integer values before the decimal point in answers I, II and III are irrelevant.

The decimal portion of the possible quotients in I, II and III are another story. From the equation we have above, for a decimal to be possible, it must be something that can be expressed as 60/y, since that is the portion of the quotient that corresponds to the decimal. But couldn't any decimal be expressed as 60 over some y? The answer is NO because we are told in the question that y is an integer.

Let's look at answer choice I first.
Does
60 = .15 , where y is an integer?
y

If we rearrange this equation, we get:
Does
60 = y , where y is an integer?
.15

This question is tantamount to asking if 60 is divisible by 0.15 or if 6000 is divisible by 15?
6000 IS divisible by 15 because it is divisible by 5 (ends in a 0) and by 3 (sum of digits, 6, is divisible by 3)
Therefore, answer choice I is CORRECT.

Using the same logic for answer choice II, we must check to see if 6000 is divisible by 16.
6000 IS divisible by 16 because it is can be divided by 2 four times: 3000, 1500, 750, 375.
Therefore, answer choice II is CORRECT.

6000 IS NOT divisible by 17 because 17 is prime and not part of the prime make-up of 6000.
Therefore answer choice III is NOT CORRECT.

Therefore the correct answer is D, I and II only.

Sorry I'm a bit late..
The best way to solve these kinda problems is by going thru the options.
Remember that number=divisor*quotient+Remainder
Option 1)
15.15Y=15Y+60; Solving we get Y as an integer.Option 1 is correct
Option 2)
18.16y=18y+60; solving we get y as an integer. Therefore, Option 2 is also correct.
Option 3)
17.17y=17y+60; solving we find that y is not an integer. Therefore, Option 3 is not correct.
Therefore correct answer is (D)

Hi Kevin,
Thanks for the question......expecting such questions more often from you.

Yea it is D ..........simple one

Hello Kedaragashe,
Glad to hear you are enjoying the questions. I post them once a week - usually on Mondays. If you want to see more, I suggest you visit our website at www.manhattangmat.com and sign up for one of our classes. They are really great and have hundreds of questions, answers and strategies to help you reach a competitive score. Our students average around 700 on the exam and our median student score is 710.
Kevin

This may be a old post but i have started my prep recently so going through all the posted questions.

This is for achal , how did you come to the equation
15.15Y=15Y+60

X/Y = Q + R
X = YQ + R

Now the options for X/Y = 15.15 which makes 15.15 = Q +60, how do we solve next to reach your step.

Thanks a lot.
Anu009

Kevin wrote:Most Manhattan GMAT students are trying to break the 700 barrier. As a result, we've developed our own math problems written at the 700+ level; these are the types of questions you'll WANT to see, when you are working at that level. Try to solve this 700+ level problem (I'll post the solution next Monday).

Question:
When the integer x is divided by the integer y, the remainder is 60. Which of the following is a possible value of the quotient x/y?

I. 15.15
II. 18.16
III. 17.17

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

The easiest approach is to understand the following:

When one positive integer doesn't divide evenly into another positive integer, we can represent what's left over as a decimal (5/2 = 2.5) or as a remainder (5/2 = 2 R 1). The problem above is testing the relationship between the decimal representation and the remainder representation. Here's the relationship:

decimal * divisor = remainder

Let's revisit 5/2 = 2.5. If we multiply the decimal (.5) by the divisor (2), we get .5 * 2 = 1, which is the remainder if we represent the division as 5/2 = 2 R1.

In the problem above, the remainder is 60, y is the divisor, and the answer choices give three possible decimal representations. For each answer choice, we should plug the given values into the formula decimal * divisor = remainder to which yields an integer value for y.

I) .15y = 60.
y = 60/.15 = 6000/15 = 400.
Since y is an integer, the correct answer must include I. Eliminate B and C.

II) .16y = 60
y = 60/.16 = 6000/16 = 375.
Since y is an integer, the correct answer must include II. Eliminate A and E.

The correct answer is D.

Got it, Thank you for explaining so well.