If x and y are positive integers, what is the remainder when x is divided by y?
(1) When x is divided by 2y, the remainder is 4
(2) When x + y is divided by y, the remainder is 4
Ans-B
Remainder-Dint quite get the statements
This topic has expert replies
-
- Master | Next Rank: 500 Posts
- Posts: 429
- Joined: Wed Sep 19, 2012 11:38 pm
- Thanked: 6 times
- Followed by:4 members
- Uva@90
- Master | Next Rank: 500 Posts
- Posts: 490
- Joined: Thu Jul 04, 2013 7:30 am
- Location: Chennai, India
- Thanked: 83 times
- Followed by:5 members
Hi Shibsriz,[email protected] wrote:If x and y are positive integers, what is the remainder when x is divided by y?
(1) When x is divided by 2y, the remainder is 4
(2) When x + y is divided by y, the remainder is 4
Ans-B
From
Statement 1: When x is divided by 2y, the remainder is 4
Let X= 10 so 2Y be 6(2*3),(Remainder will be 4)
So, X/Y = 10/3 => Remainder will be 1
Let X= 20 So 2Y be 16(2*8), (Remainder will be 4)
So, X/Y = 20/8 => Remainder will be 4
Hence Statement 1 is Insufficient.
Statement 2: When x + y is divided by y, the remainder is 4
(X+Y)/Y gives remainder 4
above statement can be written an
(X/Y) + (Y+Y) => (X/Y) + 1
Hence X/Y give remainder 4
Hence Statesmen 2 is Sufficient.
Answer is B
Regards,
Uva.
Known is a drop Unknown is an Ocean
- GMATGuruNY
- GMAT Instructor
- Posts: 15539
- Joined: Tue May 25, 2010 12:04 pm
- Location: New York, NY
- Thanked: 13060 times
- Followed by:1906 members
- GMAT Score:790
When x is divided by y, the remainder is R.[email protected] wrote:If x and y are positive integers, what is the remainder when x is divided by y?
(1) When x is divided by 2y, the remainder is 4
(2) When x + y is divided by y, the remainder is 4
This statement implies the following:
x is R more than a multiple of y.
Translated into math:
x = ky + R, where k is an integer such that k≥0.
Statement 1: When x is divided by 2y, the remainder is 4
Case 1: y=3, implying that 2y=6
Here, when x is divided by 6, the remainder is 4.
In other words, x is 4 more than a multiple of 6:
x = 6k + 4, where k is an integer such that k≥0.
Options for x = 4, 10, 16...
When these values for x are divided by y=3, we get:
x/y = 4/3 = 1 R1.
x/y = 10/3 = 3 R1.
x/y = 16/3 = 5 R1.
Result:
R=1.
Case 2: y=4, implying that 2y=8
Here, when x is divided by 8, the remainder is 4.
In other words, x is 4 more than a multiple of 8:
x = 8k + 4, where k is an integer such that k≥0.
Options for x = 4, 12, 20...
When these values for x are divided by y=4, we get:
x/y = 4/4 = 1 R0.
x/y = 12/4 = 3 R0.
x/y = 20/4 = 5 R0.
Result:
R=0.
Since x/y can yield different remainders, iNSUFFICIENT.
Statement 2: When x + y is divided by y, the remainder is 4
Case 3: y=5
Here, when x+5 is divided by 5, the remainder is 4.
In other words, x+5 is 4 more than a multiple of 5:
x + 5 = 5k + 4
x = 5k - 1, where k is an integer such that k≥1 (since x must be positive).
Options for x = 4, 9, 14...
When these values for x are divided by y=5, we get:
x/y = 4/5 = 0 R4.
x/y = 9/5 = 1 R4.
x/y = 14/5 = 2 R4.
Result:
R=4.
Case 4: y=6
Here, when x+6 is divided by 6, the remainder is 4.
In other words, x+6 is 4 more than a multiple of 6:
x + 6 = 6k + 4
x = 6k - 2, where k is an integer such that k≥1 (since x must be positive).
Options for x = 4, 10, 16...
When these values for x are divided by y=6, we get:
x/y = 4/6 = 1 R4.
x/y = 10/6 = 1 R4.
x/y = 16/6 = 2 R4.
Result:
R=4.
R=4 in both cases.
The implication is that -- in every case -- when x is divided by y, R=4.
SUFFICIENT.
The correct answer is B.
Algebraic proof for statement 2:
When x + y is divided by y, the remainder is 4.
In other words, x+y is 4 more than a multiple of y:
x+y = ky + 4.
Combining like terms, we get:
x = ky - y + 4
x = (k-1)y + 4, where k-1≥0.
Put into words:
x is 4 more than a multiple of y.
In other words:
When x is divided by y, the remainder is 4.
Last edited by GMATGuruNY on Fri Oct 25, 2013 7:39 am, edited 1 time in total.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
GMAT/MBA Expert
- Brent@GMATPrepNow
- GMAT Instructor
- Posts: 16207
- Joined: Mon Dec 08, 2008 6:26 pm
- Location: Vancouver, BC
- Thanked: 5254 times
- Followed by:1268 members
- GMAT Score:770
[email protected] wrote:If x and y are positive integers, what is the remainder when x is divided by y?
(1) When x is divided by 2y, the remainder is 4
(2) When x + y is divided by y, the remainder is 4
Target question: What is the remainder when x is divided by y?
Statement 1: When x is divided by 2y, the remainder is 4
There are several values of x and y that satisfy this condition. Here are two:
Case a: x = 4 and y = 3 (since 4 divided by 6 leaves remainder 4). Here, x divided by y leaves remainder 1
Case a: x = 4 and y = 5 (since 4 divided by 10 leaves remainder 4). Here, x divided by y leaves remainder 4
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: When x + y is divided by y, the remainder is 4
There's a nice rule that say, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Statement 2 tells us that x + y divided by y leaves remainder 4.
In other words, we can say, x + y divided by y equals some integer k with remainder 4.
So, we can write: x + y = ky + 4
Isolate x to get: x = ky - y + 4
Factor to get: x = y(k - 1) + 4
As we can see, x is 4 greater than some multiple of y.
So, if we divide x by y, the remainder must be 4
Since we can answer the target question with certainty, statement 2 is SUFFICIENT
Answer = B
Cheers,
Brent
Hi Brent,
I have a question here,
u wrote,
my doubt is
in option A) too we can write x = 2y (K) + 4 = y (2K) + 4
so, in this case too we can see, x is 4 greater than some multiple of y,
then y we are not saying statement 1 is sufficient ?
Kindly help me out..
I have a question here,
u wrote,
which i too agree,Factor to get: x = y(k - 1) + 4
As we can see, x is 4 greater than some multiple of y.
So, if we divide x by y, the remainder must be 4
Since we can answer the target question with certainty, statement 2 is SUFFICIENT
my doubt is
in option A) too we can write x = 2y (K) + 4 = y (2K) + 4
so, in this case too we can see, x is 4 greater than some multiple of y,
then y we are not saying statement 1 is sufficient ?
Kindly help me out..
-
- GMAT Instructor
- Posts: 2630
- Joined: Wed Sep 12, 2012 3:32 pm
- Location: East Bay all the way
- Thanked: 625 times
- Followed by:119 members
- GMAT Score:780
Maybe an easier way of dealing with this is to recognize that remainders are additive: in other words,
(Remainder of x divided by y) + (Remainder of y divided by y) = Remainder of (x+y) divided by y.
Now consider Statement 2. Since y has a remainder of 0 when divided by y -- as does any positive integer when divided by itself, we really have
(Remainder of x divided by y) + 0 = 4
Piece of cake!
(Remainder of x divided by y) + (Remainder of y divided by y) = Remainder of (x+y) divided by y.
Now consider Statement 2. Since y has a remainder of 0 when divided by y -- as does any positive integer when divided by itself, we really have
(Remainder of x divided by y) + 0 = 4
Piece of cake!
-
- Legendary Member
- Posts: 712
- Joined: Fri Sep 25, 2015 4:39 am
- Thanked: 14 times
- Followed by:5 members
Hi GMATGuru,GMATGuruNY wrote: Statement 2: When x + y is divided by y, the remainder is 4
Case 3: y=5
Here, when x+5 is divided by 5, the remainder is 4.
In other words, x+5 is 4 more than a multiple of 5:
x + 5 = 5k + 4
x = 5k - 1, where k is an integer such that k≥1 (since x must be positive).
Options for x = 4, 9, 14...
When these values for x are divided by y=5, we get:
x/y = 4/5 = 0 R4.
x/y = 9/5 = 1 R4.
x/y = 14/5 = 2 R4.
Result:
R=4.
Case 4: y=6
Here, when x+6 is divided by 6, the remainder is 4.
In other words, x+6 is 4 more than a multiple of 6:
x + 6 = 6k + 4
x = 6k - 2, where k is an integer such that k≥1 (since x must be positive).
Options for x = 4, 10, 16...
When these values for x are divided by y=6, we get:
x/y = 4/6 = 1 R4.
x/y = 10/6 = 1 R4.
x/y = 16/6 = 2 R4.
Result:
R=4.
R=4 in both cases.
The implication is that -- in every case -- when x is divided by y, R=4.
SUFFICIENT.
The correct answer is B.
Algebraic proof for statement 2:
When x + y is divided by y, the remainder is 4.
In other words, x+y is 4 more than a multiple of y:
x+y = ky + 4.
Combining like terms, we get:
x = ky - y + 4
x = (k-1)y + 4, where k-1≥0.
Put into words:
x is 4 more than a multiple of y.
In other words:
When x is divided by y, the remainder is 4.
I have put y=2
x+2=2k+4
x=2k+2
options for x= 4,6,10 .
in all cases Reminder is 0
I have put y=4
x+4=4k+4
x=4k
options for x= 4,8,12
In all cases reminder is 0
combined to your cases mentioned above y=5 & y=6. we have two answers R=1 & R=0. So statement 2 is insufficient.
What do I miss?
Thanks in advance
- GMATGuruNY
- GMAT Instructor
- Posts: 15539
- Joined: Tue May 25, 2010 12:04 pm
- Location: New York, NY
- Thanked: 13060 times
- Followed by:1906 members
- GMAT Score:790
RULE:
If positive integer x is divided by positive integer y, the greatest possible remainder is equal to y-1.
Example:
If y=3, the following remainders are possible:
3/3 = 1 R0.
4/3 = 1 R1.
5/3 = 1 R2.
6/3 = 2 R0.
7/3 = 2 R1.
8/3 = 2 R2.
As the red options illustrate, if y=3, the greatest possible remainder = y-1 = 3-1 = 2.
Since statement 2 indicates that the remainder is 4, y=4 is not viable.
To yield a remainder of 4, y must be GREATER THAN 4.
If positive integer x is divided by positive integer y, the greatest possible remainder is equal to y-1.
Example:
If y=3, the following remainders are possible:
3/3 = 1 R0.
4/3 = 1 R1.
5/3 = 1 R2.
6/3 = 2 R0.
7/3 = 2 R1.
8/3 = 2 R2.
As the red options illustrate, if y=3, the greatest possible remainder = y-1 = 3-1 = 2.
If y=4, then the greatest possible remainder = y-1 = 4-1 = 3.Mo2men wrote: Statement 2: When x + y is divided by y, the remainder is 4
I have put y=4
Since statement 2 indicates that the remainder is 4, y=4 is not viable.
To yield a remainder of 4, y must be GREATER THAN 4.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3