Remainder problem

BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

This topic has expert replies

Post new topic Post Reply

Previous Topic Next Topic

briology: Junior | Next Rank: 30 Posts; Posts: 22; Joined: Sun Sep 18, 2011 1:14 am

Remainder problem

by briology » Fri Nov 25, 2011 9:28 am

Would love some help with this problem. I struggle with remainder questions

.

If n is a positive integer and r is the remainder when (n-1)(n+1) is divided by 24, what is the value of r?

1. 2 is not a factor of n

2. 3 is not a factor of n

If someone can show me the best strategy for tackling this sort of problem, i'd greatly appreciate it!

Quote

Source: Beat The GMAT — Data Sufficiency | Fri Nov 25, 2011 9:28 am

neelgandham: Community Manager; Posts: 1060; Joined: Fri May 13, 2011 6:46 am; Location: Utrecht, The Netherlands; Thanked: 318 times; Followed by:52 members

by neelgandham » Fri Nov 25, 2011 9:43 am

1. 2 is not a factor of n

If n = 3, then (n-1)(n+1) = 8, remainder = 8
If n = 5, then (n-1)(n+1) = 24, remainder = 0
More than one answer, Hence insufficient!

2. 3 is not a factor of n

If n = 2, then(n-1)(n+1) = 3, remainder = 3
If n = 5, then(n-1)(n+1) = 24, remainder = 0
More than one answer, Hence insufficient!

From 1 and 2

n = 1 or 5 or 7 or 11...the remainder is 0 every time.
sufficient

Option C

Anil Gandham
Welcome to BEATtheGMAT | Photography | Getting Started | BTG Community rules | MBA Watch
Check out GMAT Prep Now's online course at https://www.gmatprepnow.com/

Quote

neelgandham: Community Manager; Posts: 1060; Joined: Fri May 13, 2011 6:46 am; Location: Utrecht, The Netherlands; Thanked: 318 times; Followed by:52 members

by neelgandham » Fri Nov 25, 2011 10:14 am

If n is a positive integer and r is the remainder when (n-1)(n+1) is divided by 24, what is the value of r?

For the benefit of fellow BTGers, here is another approach, The Algebra Way !

1. 2 is not a factor of n

Then n can be written in the form 2x+1 or 2x-1

If n = 2x+1, then(n-1)(n+1) = (2x+1-1)*(2x+1+1)= 2x*(2x+2) = 4*x*(x+1).
(n-1)*(n+1) = 4*x*(x+1), is divisible by 24 if x*(x+1) is a multiple of 6 (x=2) and is not divisible by 24 if x*(x+1) is NOT a multiple of 6 (x=4).

If n = 2x-1, then(n-1)(n+1) = (2x-1-1)*(2x-1+1)= 2x*(2x-2) = 4*x*(x-1).
(n-1)*(n+1) = 4*x*(x-1), is divisible by 24 if x*(x-1) is a multiple of 6 (x=3) and is not divisible by 24 if x*(x-1) is NOT a multiple of 6 (x=2).

Hence Insufficient!

2. 3 is not a factor of n

Then n can be written in the form 3x+1, 3x-1, 3x+2, or 3x-2

If n = 3x+1, then(n-1)(n+1) = (3x+1-1)*(3x+1+1)= 3x*(3x+2)
(n-1)*(n+1) = 3x*(3x+2), is divisible by 24 if x*(3x+2) is a multiple of 8 (x=2) and is not divisible by 24 if x*(3x+2) is NOT a multiple of 8 (x=3).

If n = 3x-1, then(n-1)(n+1) = (3x-1-1)*(3x-1+1)= 3x*(3x-2)
(n-1)*(n+1) = 3x*(3x-2), is divisible by 24 if x*(3x-2) is a multiple of 8 (x=2) and is not divisible by 24 if x*(3x-2) is NOT a multiple of 8 (x=3).

If n = 3x+2, then(n-1)(n+1) = (3x+2-1)*(3x+2+1)= (3x+1)*(3x+3) = 3*(x+1)*(3x+1)
(n-1)*(n+1) = 3*(x+1)*(3x+1), is divisible by 24 if (x+1)*(3x+1) is a multiple of 8 (x=3) and is not divisible by 24 if (x+1)*(3x+1) is NOT a multiple of 8 (x=4).

If n = 3x-2, then(n-1)(n+1) = (3x-2-1)*(3x-2+1)= (3x-1)*(3x-3) = 3*(x-1)*(3x-1)
(n-1)*(n+1) = 3*(x-1)*(3x-1), is divisible by 24 if (x-1)*(3x-1) is a multiple of 8 (x=3) and is not divisible by 24 if (x-1)*(3x-1) is NOT a multiple of 8 (x=4).

From 1 and 2

n can be written in the form 6x+1(n>=0) or 6x-1(n>0)

If n = 6x+1, then(n-1)(n+1) = (6x+1-1)*(6x+1+1)= 6x*(6x+2) = 12*x*(3x+1).
If x is even, 3x+1 = odd and 12*x*(3x+1) = 12*even*odd = 24*An Integer. Hence the value of r = 0
If x is odd, 3x+1 = even and 12*x*(3x+1) = 12*odd*even = 24*An Integer. Hence the value of r = 0

If n = 6x-1, then(n-1)(n+1) = (6x-1-1)*(6x-1+1)= 6x*(6x-2) = 12*x*(3x-1).
If x is even, 3x-1 = odd and 12*x*(3x-1) = 12*even*odd = 24*An Integer. Hence the value of r = 0
If x is odd, 3x-1 = even and 12*x*(3x-1) = 12*odd*even = 24*An Integer. Hence the value of r = 0

Hence Sufficient, Answer C
Please correct me if I am wrong !

Anil Gandham
Welcome to BEATtheGMAT | Photography | Getting Started | BTG Community rules | MBA Watch
Check out GMAT Prep Now's online course at https://www.gmatprepnow.com/

Quote

sk8legend408: Senior | Next Rank: 100 Posts; Posts: 62; Joined: Fri Aug 12, 2011 12:37 am; Thanked: 3 times; Followed by:1 members

by sk8legend408 » Sat Nov 26, 2011 8:41 am

Hey guys I like your approaches but was just wondering once we start testing whether both statements together are sufficient, doesn't the remainder change if we use prime numbers greater than 24?

If that is the case then the answer would be E.

Experts would love to hear your thoughts.

Quote

zooki: Junior | Next Rank: 30 Posts; Posts: 29; Joined: Fri Oct 14, 2011 10:15 am; Thanked: 4 times; Followed by:1 members

by zooki » Sat Nov 26, 2011 11:53 am

The quest becomes: what is remainder of (n^2-1)/(2^3*3)

statement 1: 2 is not a factor of n. i. e. n is odd number >2, and n^2-1= even. Not Enough.
statement 2: 3 is not a factor of n. i. e. n is odd number >3, and n^2-1= even. Not Enough.
Statement 1+ 2: n is prime number >3: now try for 5, 7, and 11. The remainder for (n^2-1)/(2^3*3) is same.

Quote

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

by GMATGuruNY » Sat Nov 26, 2011 3:37 pm

briology wrote:Would love some help with this problem. I struggle with remainder questions .

If n is a positive integer and r is the remainder when (n-1)(n+1) is divided by 24, what is the value of r?

1. 2 is not a factor of n

2. 3 is not a factor of n

If someone can show me the best strategy for tackling this sort of problem, i'd greatly appreciate it!

Statement 1: 2 is not a factor of n.
Thus, n = odd.
Thus, (n-1)(n+1) = the product of two consecutive even integers.
Of every two consecutive even integers, exactly one is a multiple of 4.
Thus, the product of 2 consecutive even integers = the product of an even integer and a multiple of 4 = a multiple of 8.
Since a multiple of 8 can be a multiple of 24 (in which case r=0) or not be a multiple of 24 (in which case râ‰ 0), INSUFFICIENT.

Statement 2: 3 is not a factor of n
Since one of every 3 consecutive integers is a multiple of 3, and n is not a multiple of 3, either (n-1) or (n+1) must be a multiple of 3.
Thus, (n-1)(n+1) = a multiple of 3.
If (n-1)(n+1) is also a multiple of 8, then (n-1)(n+1) = a multiple of 24, in which case r=0.
If (n-1)(n+1) is not a multiple of 8, then (n-1)(n+1) â‰ a multiple of 24, in which case râ‰ 0.
INSUFFICIENT.

Statements 1 and 2 combined:
Since (n-1)(n+1) = a multiple of 8, and either n-1 or n+1 must be a multiple of 3, (n-1)(n+1) = a multiple of 24.
When a multiple of 24 is divided by 24, r=0.
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

Quote

amit2k9: Master | Next Rank: 500 Posts; Posts: 461; Joined: Tue May 10, 2011 9:09 am; Location: pune; Thanked: 36 times; Followed by:3 members

by amit2k9 » Sun Nov 27, 2011 9:51 pm

n = 1,5,7
all cases r=0
hence C

For Understanding Sustainability,Green Businesses and Social Entrepreneurship visit -https://aamthoughts.blocked/
(Featured Best Green Site Worldwide-https://bloggers.com/green/popular/page2)