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# When the positive integer x is divided by the positive integ

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Manager
Joined: 17 Oct 2012
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Location: India
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When the positive integer x is divided by the positive integ [#permalink]

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09 Jul 2014, 08:01
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Difficulty:

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Question Stats:

61% (01:54) correct 39% (02:38) wrong based on 149 sessions

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When the positive integer x is divided by the positive integer y, the quotient is 2 and the remainder is z. When x is divided by the positive integer a, the quotient is 3 and the remainder is b. Is z > b?

(1) The ratio of y to a is less than 3 to 2.
(2) The ratio of y to a is greater than 2 to 3.
[Reveal] Spoiler: OA
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Re: When the positive integer x is divided by the positive integ [#permalink]

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09 Jul 2014, 08:39
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chetan86 wrote:
When the positive integer x is divided by the positive integer y, the quotient is 2 and the remainder is z. When x is divided by the positive integer a, the quotient is 3 and the remainder is b. Is z > b?

(1) The ratio of y to a is less than 3 to 2.
(2) The ratio of y to a is greater than 2 to 3.

Given :
x=2y+z
x=3a+b

Therefor, 2y+z=3a+b

Statement 1:
y/a<3/2
2y<3a

since,2y+z=3a+b and 2y<3a, z>b for equality to hold true.
sufficient statement

Statement 2:
y/a>2/3
3y>2a

Statement is insufficient to comment whether z>b.

Ans=A
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Joined: 21 Jun 2014
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When the positive integer x is divided by the positive integ [#permalink]

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Updated on: 09 Jul 2014, 21:45
Curious to know whether z or b can be negative. In general, do we have to consider the case for negative remainders as well in such questions
Please shed some light on this.

Originally posted by jogeshanand on 09 Jul 2014, 10:23.
Last edited by jogeshanand on 09 Jul 2014, 21:45, edited 1 time in total.
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Posts: 44600
Re: When the positive integer x is divided by the positive integ [#permalink]

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09 Jul 2014, 10:36
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Expert's post
jogeshanand wrote:
Curious to know whether z and b can be negative. In general, do we have to consider the case for negative remainders as well in such questions
Please shed some light on this.

No, at least on the GMAT.

OG12:
If $$x$$ and $$y$$ are positive integers, there exist unique integers $$q$$ and $$r$$, called the quotient and remainder, respectively, such that $$y =divisor*quotient+remainder= xq + r$$ and $$0\leq{r}<x$$.

Theory on remainders problems: remainders-144665.html
Tips: tips-and-hints-for-specific-quant-topics-with-examples-172096.html#p1376126

Units digits, exponents, remainders problems: new-units-digits-exponents-remainders-problems-168569.html

All DS remainders problems to practice: search.php?search_id=tag&tag_id=198
All PS remainders problems to practice: search.php?search_id=tag&tag_id=199

Hope this helps.
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Re: When the positive integer x is divided by the positive integ [#permalink]

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07 Mar 2017, 13:53
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Posting official solution of this problem.
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15 Apr 2018, 04:46
When the positive integer x is divided by the positive integer y the quotient is 2 and the remainder is z. when x is divided by the positive integer a the quotient is 3 and the remainder is b. Is z>b ?

1)the ration of y to a is less than 3 to 2.
2)the ratio of y to a is greater than 2 to 3.
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15 Apr 2018, 05:16
selim wrote:
When the positive integer x is divided by the positive integer y the quotient is 2 and the remainder is z. when x is divided by the positive integer a the quotient is 3 and the remainder is b. Is z>b ?

1)the ration of y to a is less than 3 to 2.
2)the ratio of y to a is greater than 2 to 3.

When the positive integer x is divided by the positive integer y the quotient is 2 and the remainder is z

i.e. x = 2y + z

when x is divided by the positive integer a the quotient is 3 and the remainder is b

i.e. x = 3a + b

i.e. 2y + z = 3a + b

Question : Is z > b ?
Or
Question : Is 2y > 3a ?

Statement 1: the ratio of y to a is less than 3 to 2.

i.e. y / a < 3/2
i.e. 2y < 3a
SUFFICIENT

Statement 2: the ratio of y to a is greater than 2 to 3

i.e. y / a > 2/3
i.e. 3y < 2a
NOT SUFFICIENT

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15 Apr 2018, 07:57
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GMATinsight wrote:
selim wrote:
When the positive integer x is divided by the positive integer y the quotient is 2 and the remainder is z. when x is divided by the positive integer a the quotient is 3 and the remainder is b. Is z>b ?

1)the ration of y to a is less than 3 to 2.
2)the ratio of y to a is greater than 2 to 3.

When the positive integer x is divided by the positive integer y the quotient is 2 and the remainder is z

i.e. x = 2y + z

when x is divided by the positive integer a the quotient is 3 and the remainder is b

i.e. x = 3a + b

i.e. 2y + z = 3a + b

Question : Is z > b ?
Or
Question : Is 2y > 3a ?

Statement 1: the ratio of y to a is less than 3 to 2.

i.e. y / a < 3/2
i.e. 2y < 3a
SUFFICIENT

Statement 2: the ratio of y to a is greater than 2 to 3

i.e. y / a > 2/3
i.e. 3y < 2a
NOT SUFFICIENT

Dear GMATinsight

While I have arrived to same conclusion for statement 1 as you did, I have arrived to different conclusion for statement 2.

Statement 2: the ratio of y to a is greater than 2 to 3

i.e. y / a > 2/3
i.e. 3y < 2a

We are dealing with POSITIVE integers.....so 3a > 2a & 3y > 2y
But we have 2a > 3y Therefore:

3a > 2a > 3y > 2y......3a > 2y...same as statement 1...........Sufficient

Another way to plug positive integer numbers:

Let a =2 & y = 1.........2 (2) > 3(1)...... 2y + z = 3a + b..........2+z = 6+b.............z-b = 4...z must be greater than b

Let a =3 & y = 1.........2 (3) > 3(1)...... 2y + z = 3a + b..........2+z = 9+b.............z-b = 7...z must be greater than b

Let a =7 & y = 6.........2 (7) > 3(6)...... 2y + z = 3a + b..........12+z = 21+b.............z-b = 8...z must be greater than b

Sufficient

Where did I wrong in my solution?

Thanks
Math Expert
Joined: 02 Sep 2009
Posts: 44600
Re: When the positive integer x is divided by the positive integ [#permalink]

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15 Apr 2018, 09:21
selim wrote:
When the positive integer x is divided by the positive integer y the quotient is 2 and the remainder is z. when x is divided by the positive integer a the quotient is 3 and the remainder is b. Is z>b ?

1)the ration of y to a is less than 3 to 2.
2)the ratio of y to a is greater than 2 to 3.

Please follow the rules when posting a question: https://gmatclub.com/forum/rules-for-po ... 33935.html Pay attention to rules 1 and 3. Thank you.
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Re: When the positive integer x is divided by the positive integ [#permalink]

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15 Apr 2018, 11:03
Mo2men wrote:
GMATinsight wrote:
selim wrote:
When the positive integer x is divided by the positive integer y the quotient is 2 and the remainder is z. when x is divided by the positive integer a the quotient is 3 and the remainder is b. Is z>b ?

1)the ration of y to a is less than 3 to 2.
2)the ratio of y to a is greater than 2 to 3.

When the positive integer x is divided by the positive integer y the quotient is 2 and the remainder is z

i.e. x = 2y + z

when x is divided by the positive integer a the quotient is 3 and the remainder is b

i.e. x = 3a + b

i.e. 2y + z = 3a + b

Question : Is z > b ?
Or
Question : Is 2y > 3a ?

Statement 1: the ratio of y to a is less than 3 to 2.

i.e. y / a < 3/2
i.e. 2y < 3a
SUFFICIENT

Statement 2: the ratio of y to a is greater than 2 to 3

i.e. y / a > 2/3
i.e. 3y < 2a
NOT SUFFICIENT

Dear GMATinsight

While I have arrived to same conclusion for statement 1 as you did, I have arrived to different conclusion for statement 2.

Statement 2: the ratio of y to a is greater than 2 to 3

i.e. y / a > 2/3
i.e. 3y < 2a

We are dealing with POSITIVE integers.....so 3a > 2a & 3y > 2y
But we have 2a > 3y Therefore:

3a > 2a > 3y > 2y......3a > 2y...same as statement 1...........Sufficient

Another way to plug positive integer numbers:

Let a =2 & y = 1.........2 (2) > 3(1)...... 2y + z = 3a + b..........2+z = 6+b.............z-b = 4...z must be greater than b

Let a =3 & y = 1.........2 (3) > 3(1)...... 2y + z = 3a + b..........2+z = 9+b.............z-b = 7...z must be greater than b

Let a =7 & y = 6.........2 (7) > 3(6)...... 2y + z = 3a + b..........12+z = 21+b.............z-b = 8...z must be greater than b

Sufficient

Where did I wrong in my solution?

Thanks

Hello

y/a > 2/3 means 3y > 2a. I think by mistake you have written it as 3y < 2a
Re: When the positive integer x is divided by the positive integ   [#permalink] 15 Apr 2018, 11:03
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