If x and y are different positive integers, which of the following COU

Question

If x and y are different positive integers, which of the following COULD be true: 
i) When x is divided by y, the remainder is x
ii) When 2x is divided by y, the remainder is x
iii) When x+y is divided by x , the remainder is x-y

A) i only
B) i & ii only
C) i & iii only
D) ii & iii only
E) i, ii & iii

ASIDE: Many Integer Properties questions can be solved by identifying values that satisfy some given conditions. This question is intended to strengthen that skill.

KSBGC · Answer

First of all this question requires could be true answer.

i) could be.

This statement indicates that x has to be less than y.

x = 3

y = 4

3/4............3 is remainder here. Here the definition of remainder is needed.

So could be true.

ii) could be

iii) x = 2 and y = 1

2 + 1 / 2 = 2 *1 + 1

Remainder is 1

x - y

= 2 - 1 = 1.

Could be .

Actually there is no relationship between x and y is given. Thus any of them could be true in different ways.

The best answer is E.

BrentGMATPrepNow · Answer

Hi  selim ,

There's an error in one of your calculations.

Cheers,
Brent

KSBGC · Answer

got it . thanks .

varun4s · Answer

All the options are possible as there is no exact relationship between integer x and y. So E is the answer.

chetan2u · Answer

Great question !! Brent..
let us see the question

i) When x is divided by y, the remainder is x
 Whenever y>x, it will be true 
example y=7 x=3, remainder is x or 3..possible

ii) When 2x is divided by y, the remainder is x
so let us make the equation..
 2x=ny+x......ny=x , so x is multiple of y and   remainder   will always be 0... NOT possible

iii) When x+y is divided by x , the remainder is x-y
 when x=y is divided by x, remainder will be y 
Now can this y be equal to x-y....y=x-y... x=2y
so possible.. let y = 5, x=10.. remainder = 10-5=5=x-y....
so possible

(i) and (iii)

C

rahitgs · Answer

Hi Chetan, could you spare a few minutes to explain iii again? Thx

Mansoor50 · Answer

Great question !! Brent..
let us see the question

i) When x is divided by y, the remainder is x
 Whenever y>x, it will be true 
example y=7 x=3, remainder is x or 3..possible

ii) When 2x is divided by y, the remainder is x
so let us make the equation..
 2x=ny+x......ny=x , so x is multiple of y and   remainder   will always be 0... NOT possible

iii) When x+y is divided by x , the remainder is x-y
 when x=y is divided by x, remainder will be y 
Now can this y be equal to x-y....y=x-y... x=2y
so possible.. let y = 5, x=10.. remainder = 10-5=5=x-y....
so possible

(i) and (iii)

C

Hi chetan...

I dont understand your algebra for the III. can you elaborate please?

regards

hibobotamuss · Answer

Is zero considered a positive integer? Because then X=0 would hold true for II, no?

BrentGMATPrepNow · Answer

Zero is neither positive no negative.

Cheers,
Brent

BrentGMATPrepNow · Answer

If all of the statements are true, then for each statement, you should be able to specify values for x and y that satisfy the statement.

Hint: This is not possible for each statement (i.e., the correct answer is not E)

Cheers,
Brent

Mansoor50 · Answer

Algebraic approach to (iii)

x+y divided by x gives remainder of x-y

x+y-->px + x -y
2y-->px, where p is a positive integer. this means:
2y can be: x, 2x, 3x, 4x, ,,, and so on

2y eq x means y is x/2,,,,picking x as 4 gives y as 2 and thus
6/4 gives a remainder of 2 which is 4-2--> x-y

therefore, possible

jorgetomas9 · Answer

Excuse me, how dividing 3/7 can give you a remainder of 3?

BrentGMATPrepNow · Answer

A few examples to set the mood: 
25 divided by 8 equals 3 with remainder 1. From this result, we can write (3)(8) + 1 = 25
17 divided by 3 equals 5 with remainder 2. From this result, we can write (5)(3) + 2 = 17
64 divided by 10 equals 6 with remainder 4. From this result, we can write (6)(10) + 4 = 64
And now.....

3 divided by 7 equals 0 with remainder 3. From this result, we can write (0)(7) + 3 = 3

Cheers,
Brent

BrentGMATPrepNow · Answer

i) When x is divided by y, the remainder is x 
This occurs any time x < y
For example, if x = 5 and y = 7, then statement i becomes: When 5 is divided by 7, the remainder is 5
So true!

Scan the answer choices....eliminate D
-----------------------------------------------------
 ii) When 2x is divided by y, the remainder is x 
Nice rule:   If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.    
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

So, from statement ii, we can say: some possible values of 2x are: x, x + y, x + 2y, x + 3y, . . . etc
Let's examine the first option: 2x = x. Solve to get x = 0, but we're told x is POSITIVE No good.

Check the second option: 2x = x + y. Solve to get x = y. This means the remainder is y (aka x), but the remainder CANNOT be greater than the divisor. See the rule below:

When positive integer N is divided by positive integer D, the remainder R is such that 0 ≤ R < D  
For example, if we divide some positive integer by 7, the remainder will be 6, 5, 4, 3, 2, 1, or 0

Check the third option: 2x = x + 2y. Solve to get x = 2y. 
This means the remainder = 2y, which means the remainder is greater than the divisor (see rule above). No good.

In fact, we can see that, with all of the possible values of 2x, the remainder will be greater than the divisor. 
So, statement ii is NOT true.

Scan the answer choices....eliminate B and E
-----------------------------------------------------
 iii) When x+y is divided by x , the remainder is x-y 
Some possible values of x+y are: (x-y), (x-y)+x, (x-y)+2x, (x-y)+3x, . . . etc
Let's examine the first option: x+y = x-y
Solve to get y = 0. No good.

Check the second option: x+y = (x-y)+x
Simplify: x+y = 2x - y
Solve to get: x = 2y

So, one possible case is: x = 6 and y = 3
Statement iii becomes: When (6 + 3) is divided by 6, the remainder is 3
So true!
------------------------------

Answer: C

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