Can someone please advise on how to solve this algebraically?
If m>n, then is mn divisible by 3?
(1) The remainder when m + n is divided by 6 is 5
(2) The remainder when m - n is divided by 6 is 3
OA is C
Or if number picking is the only option, how to pick smart numbers?
Thanks!
Integer Properties
This topic has expert replies
-
- Master | Next Rank: 500 Posts
- Posts: 103
- Joined: Sat Jun 02, 2012 9:46 pm
- Thanked: 1 times
- 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
Statement 1:topspin360 wrote:Can someone please advise on how to solve this algebraically?
If m>n, then is mn divisible by 3?
(1) The remainder when m + n is divided by 6 is 5
(2) The remainder when m - n is divided by 6 is 3
OA is C
In other words, m+n is equal to 5 more than a multiple of 6:
m+n = 5, 11, 17, 23, 29...
If m+n=5, the following cases are possible:
Case 1:m=3 and n=2
In this case, mn=6, which is divisible by 3.
Case 2: m=4 and n=1
In this case, mn=4, which is not divisible by 3.
INSUFFICIENT.
Statement 2:
In other words, m-n is equal to 3 more than a multiple of 6:
m-n = 3, 9, 15, 21, 27...
If m-n=3, the following cases are possible:
Case 1: m=6 and n=3
In this case, mn=18, which is divisible by 3.
Case 2: m=5 and n=2
In this case, mn=10, which is not divisible by 3.
INSUFFICIENT.
Statements combined:
Case 1: m+n=5 and m-n=3
Adding the equations, we get:
2m=8
m=4, implying that n=1.
In this case, mn=4, which is not divisible by 3.
Case 2: m+n=23 and m-n=9
Adding the equations, we get:
2m=32
m=16, implying that n=7
In this case, mn=16*7, which is not divisible by 3.
In each case, mn is not divisible by 3.
Maybe one more case to be safe.
Case 3: m+n=29 and m-n=21
Adding the equations, we get:
2m=50
m=25, implying that n=4.
In this case, mn=25*4, which is not divisible by 3.
The 3 random cases above illustrate that -- when the statements are combined -- mn in every case will not be divisible by 3.
SUFFICIENT.
The correct answer is C.
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
For mn to be divisible by 3, we should have either one of two as a multiple of 3
Statement#1
m+n = 6x+5................1
Insufficient by itself
Statement#2
m-n = 6y-3.............2
Insufficient by itself
Combining statement 1 & 2,
Using equation 1&2 and solving for m we get
m= 3x+3y+1
where, 3x is multiple of 3, 3y is multiple of 3 but with addition of 1 at the end whole sum is NOT multiple of 3, hence m is not multiple of 3
Again using equation 1 & 2 and solving for n we get,
n=3x-3y+4
Using same logic as above, we can deduce that n is not a multiple of 3
And since both M and N are not multiple of 3, MN can not be multiple of 3
Combining 1 & 2 Sufficient
Statement#1
m+n = 6x+5................1
Insufficient by itself
Statement#2
m-n = 6y-3.............2
Insufficient by itself
Combining statement 1 & 2,
Using equation 1&2 and solving for m we get
m= 3x+3y+1
where, 3x is multiple of 3, 3y is multiple of 3 but with addition of 1 at the end whole sum is NOT multiple of 3, hence m is not multiple of 3
Again using equation 1 & 2 and solving for n we get,
n=3x-3y+4
Using same logic as above, we can deduce that n is not a multiple of 3
And since both M and N are not multiple of 3, MN can not be multiple of 3
Combining 1 & 2 Sufficient
-
- Master | Next Rank: 500 Posts
- Posts: 103
- Joined: Sat Jun 02, 2012 9:46 pm
- Thanked: 1 times