When the integer x is divided by the integer y

"x is divided by the integer y, the remainder is 60" --> \(x=qy+60\) --> or \(\frac{x}{y}=q+\frac{60}{y}\), so basically \(\frac{60}{y}\) is a decimal portion of \(\frac{x}{y}\).

For example: \(\frac{x}{y}=15.15\) means that \(\frac{x}{y}=15+0.15=15+\frac{60}{y}\)

Let's look at the options:

I. If \(\frac{x}{y}=15.15\) then \(\frac{60}{y}=0.15\) and \(y=400=integer\), which is possible;

II. If \(\frac{x}{y}=18.16\) then \(\frac{60}{y}=0.16\) and \(y=375=integer\), which is possible;

III. If \(\frac{x}{y}=17.17\) then \(\frac{60}{y}=0.17\) and \(y=\frac{6000}{17}\neq{integer}\), which is not possible as \(y\) must be an integer.

Answer: D.
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In x=yk+60, y = divisor and k = quotient.

On dividing both sides of the equation by Y --------> \(\frac{x}{y} = \frac{(yk)}{y} + \frac{60}{y}\) --------> \(\frac{x}{y} = k + \frac{60}{y}\)

So k = quotient and \(\frac{60}{y}\)is Remainder. -------- Quotient can not be fraction.

So while considering 15.15 we should recognize that 15 must be quotient and 0.15 must be remainder, but the remainder is \(\frac{60}{y}\)---------> Hence \(0.15 = \frac{60}{y}\) -------> \(Y = \frac{60}{0.15}\) -------> Y = 400

Note that we are told that Y is integer so basically we are just looking for whether given expression could give the integer value of Y

Consider second option 18.16 ------> Y = 60/0.16 -------> 6000/16 ------> some integer

In 17.17 ---------> 60/0.17 -------> 6000/17 ------> Not an Integer

Hope that helps

First check this video that discusses the decimal representation of remainders: https://youtu.be/A5abKfUBFSc

The decimal part gives us the possible remainders and the divisors.

\(I. 15.15 = 15 + 0.15 = 15 + \frac{15}{100}=15 + \frac{3}{20} \)

So the remainder can be any multiple of 3 here and the corresponding divisor will be the same multiple of 20. Hence the remainder can be 60 here since it is a multiple of 3.

\(II. 18.16 = 18 + \frac{16}{100}=18 + \frac{4}{25}\)

So the remainder can be any multiple of 4 here and the corresponding divisor will be the same multiple of 25. Hence the remainder can be 60 here since it is a multiple of 4.

\(III. 17.17 = 17 + \frac{17}{100}\)

Since 60 is not a multiple of 17, the remainder cannot be 60 here.

Answer (D)
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Is the second option 18.18 ?

No, it's 18.16. There was a typo in solution.
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Bunuel,

I am confused by the terminology of the question. When x is divided by y the remainder 60. is x=yk+60. "y" is the quotient and the question is asking for possible quotients, ie, possible y values.

But in your solution you have x/y=quotient. Very confused.

Hello!

Would it be possible to solve the task as follows:

I. We know that the reminder is 15. As 60 is divisible by 15 we can assume that 15.15 could be one solution.
II. We know that the reminder is 16 which is 2*8. As 60 can be divided by 2 we can assume that 18.16 could be one solution.
III. We know that the reminder is 17. As 60 is not divisible by 17 or vice versa, we can assume that 17.17 cannot be one of the solutions.

Also, please explain if my approach has a flaw.

Thank you in advance!

I think, it will not work every time. I will show you how.

Let's change the third option. Lets assume Remainder is 21 and quotient is 5.10

We know that the remainder is 0.10 As 21 is not divisible by 10 or vice versa, we can assume that 5.10 cannot be one of the solutions. Can this be true??
21/0.10 ------> 2100 / 10 ---------> well divisible by 10.

in the division 21/0.10 when you move the decimal sign of denominator to the right by two places, you add two zeros in the numerator to create the fraction 2100/10
When you are awarding two zeros to the numerator (i.e. to 21) remember you are actually giving 2 twos and 2 fives to the numerator. Why is so. That is because One zero can be obtained by multiplying 5 with 2
Now since the denominator (i.e. 10) contains 1 five and 1 two and since you just have given 2 fives and 2 twos, you can surely divide numerator by the denominator.

Thank you very much for you explanation!

The question is simply asking IF the decimal part of the values given in I, II & III yield 60 when multiplied by y, while keeping y as integer.

Let me show how,

I. 15.15 ===> .15*y=60 is y integer? yes, y= 6000/15 = 400 hence 15.15 can be quotient.
II 18.16 ===> .16*y=60 is y integer? yes, y= 6000/16 = 375 hence 18.16 can be quotient.
III 17.17 ===> .17*y=60 is y integer? NO, y=6000/17 hence 17.17 can not be quotient.

I and II Only.

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 was totally confused...Although the source is MGMAT.. I don't agree with the terminology..
The question demands that we find the value of the quotient of X/Y...i.e the value of Green colored portion in X/Y= Q + 60/Y
But the question implies us to find out the value of X/Y i.e Q+60/Y
Consider the case: 7=1.2*5 + 1...Although this is unreasonable..but this is what I took from the Question

[quote="enigma123"]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


I don't have a clue how this will be solved. Can someone please help? Unfortunately, I don't have an OA either.[/quo


It is better to evaluate possible quotients as integer + fraction manner (decimal part), exploiting idea that decimals come after dividing remainder to divisor, so:

I. 15.15. is 15+15/100 where 15 is factor of 60, so it can be quotient

II. 18.16 is 18+16/100 or 4/25 where 4 is a factor of 60, so it can be quotient

III. 17.17 is 17+17/100 where 17 is not a factor of 60, so it cannot be quotient

Answer is D

Many seemingly difficult remainder problems can be simplified using the following concept:
If you take the decimal portion of the resulting number when you divide ''x' by "y", and multiply it to "y", you will get the remainder.

Let's look at the options:

I. 15.15 ===> .15*y=60 is y integer? yes, y= 6000/15 = 400 hence 15.15 can be quotient.
II 18.16 ===> .16*y=60 is y integer? yes, y= 6000/16 = 375 hence 18.16 can be quotient.
III 17.17 ===> .17*y=60 is y integer? NO, y=6000/17 hence 17.17 can not be quotient.

Answer: D

Problem statement was difficult to parse. but some how I managed to understand what we can do.

1) x/y = 15 + 0.15
2) x/y = 18 + 0.16
3) x/y = 17 + 0.17

w.k.t x = yn (i.e Q) + yk=60 (i.e remainder)
therefore :
1) y*0.15=60 -- we can get y an integer.
2) y*0.16=60 -- we can get y an integer.
3) y*0.17=60 -- we can't get y an integer.

Therefore 1 and 2 are possible answers.
Ans D.

solved this in 2 minutes , must be getting better at integers.

Followed Bunuel's approach

put each 1,2,3 into the form

60/y = decimal value

i) and ii) gave an integer for y, there fore good,

iii) did not, therefore X

d)

first of all..rebuild the formula:
x/y=quotient+remainder
x=q*y + r
r=x-q*y

1. 15.15
q=15.
r = 60/y = 15/100
simplify 60 with 15 = 4/y=1/100
y=400
so it's possible.
x=15*400+60
note that 60/400 = 0.15

i'll start with 2, since the numbers are even and are easier to work with..
q=18, and r=60/y=16/100
simplify 60 with 4, and 16 with 4, then 4 and 100 with 4.
15/y=1/25
y=375.

again, it is possible.
60/375 = 0.16


so far, we could have eliminated all but D.
but let's test for III.


q=17, r=60/y = 17/100
17y=6000
6000 is not divisible by 17.
but we are told that y is an integer.

thus, III is not true.

Excellent Question..
here is what we need to know prior to solving this one=>
When a positive integer X is Divided by a Positive integer Y => the decimal part in the quotient => Remainder
Hence In all the cases the quotient is viable only if the remainder which is 60 can be divided by the decimal part.
now as i one and 2 .15 and .16 both divide 60 but .17 does not Hence D is the answer.
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Letting Q = the quotient, we can use the expression:

x/y = Q + 60/y

We see that 60/y is the remainder. We can set this to the decimal portion of each value in the Roman numerals and solve for y.

I. 15.15

60/y = 0.15

60 = 0.15y

y = 60/0.15 = 6000/15 = 400

We see that 15.15 could be a value of x/y since we have y as an integer.

II. 18.16

60/y = 0.16

60 = 0.16y

y = 60/0.16 = 6000/16 = 375

We see that 18.16 could be a value of x/y since we have y as an integer.

III. 17.17

60/y = 0.17

60 = 0.17y

y = 60/0.17 = 6000/17 = 352.94

We see that 17.17 could NOT be a value of x/y since we DON’T have y as an integer.

Answer: D
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I'm very confused with the wording of the question. What does it mean by "..possible value of the quotient x/y"?
Does it mean that x/y is the quotient or that we have to find the value of the quotient of x/y when 60 is the remainder? Bunuel please help