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

Redeem

Reminders

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
Master | Next Rank: 500 Posts
Posts: 468
Joined: Mon Jul 25, 2011 10:20 pm
Thanked: 29 times
Followed by:4 members

Reminders

by vipulgoyal » Thu Aug 07, 2014 1:06 am
What is the remainder when X^4 + Y^4 is divided by 5

1. When X-Y is divided by 5 remainder is 1
2. When X+Y is divided by 5 remainder is 2

C
Top
Source: — Data Sufficiency |
User avatar
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 » Thu Aug 07, 2014 3:02 am
vipulgoyal wrote:What is the remainder when X^4 + Y^4 is divided by 5

1. When X-Y is divided by 5 remainder is 1
2. When X+Y is divided by 5 remainder is 2

C
When integer m is divided by 5, the remainder depends on the UNITS digit of m.
If the units digit of m is 0 or 5, dividing by 5 will yield a remainder of 0.
If the units digit of m is 1 or 6, dividing by 5 will yield a remainder of 1.
If the units digit of m is 2 or 7, dividing by 5 will yield a remainder of 2.
If the units digit of m is 3 or 8, dividing by 5 will yield a remainder of 3.
If the units digit of m is 4 or 9, dividing by 5 will yield a remainder of 4.

Statement 1:
In other words, x-y is a (MULTIPLE OF 5) + 1.
Thus:
x-y = 5a + 1 = 1, 6, 11, 16, 21...

Let x-y = 1.
If x=1 and y=0, then the units digit of x� + y� is 1, in which case dividing by 5 will yield a remainder of 1.
If x=2 and y=1, then the units digit of x� + y� is 7, in which case dividing by 5 will yield a remainder of 2.
Since the remainder can be different values, INSUFFICIENT.

Statement 2:
In other words, x+y is a (MULTIPLE OF 5) + 2.
Thus:
x+y = 5b + 2 = 2, 7, 12, 17, 22...

Let x+y = 2.
If x=2 and y=0, then the units digit of x� + y� is 6, in which case dividing by 5 will yield a remainder of 1.
If x=1 and y=1, then the units digit of x� + y� is 2, in which case dividing by 5 will yield a remainder of 2.
Since the remainder can be different values, INSUFFICIENT.

Statements combined:
Statement 2: x+y = 2, 7, 12, 17, 22...
Statement 1: x-y = 1, 6, 11, 16, 21...

If x+y=7 and x-y=1, then x=4 and y=3.
Here, the units digit of x� + y� is 7, in which case dividing by 5 will yield a remainder of 2.
If x+y=12 and x-y=6, then x=9 and y=3.
Here, the units digit of x� + y� is 2, in which case dividing by 5 will yield a remainder of 2.
If x+y=17 and x-y=1, then x=9 and y=8.
Here, the units digit of x� + y� is 7, in which case dividing by 5 will yield a remainder of 2.

In every case, dividing x� + y� by 5 yields a remainder of 2.
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
Top
User avatar
Legendary Member
Posts: 1100
Joined: Sat May 10, 2014 11:34 pm
Location: New Delhi, India
Thanked: 205 times
Followed by:24 members

by GMATinsight » Thu Aug 07, 2014 6:26 am
vipulgoyal wrote:What is the remainder when X^4 + Y^4 is divided by 5

1. When X-Y is divided by 5 remainder is 1
2. When X+Y is divided by 5 remainder is 2

C
Question : Remainder when X^4 + Y^4 is divided by 5 ?

Statement 1) X - Y = 5a+1

i.e. X - Y = 6 or 11 or 16 and so on...
Case 1: X-Y = 6 i.e. X = 7, Y = 1 then X^4 + Y^4 = 7^4 + 1^4 = Unit digit (1+1) = Unit digit 2
i.e. Remainder when divided by 5 = 2

Case 2: X-Y = 6 i.e. X = 15, Y = 9 then X^4 + Y^4 = 5^4 + 9^4 = Unit digit (5+1) = Unit digit 6
i.e. Remainder when divided by 5 = 1
INCONSISTENT SOLUTIONS

INSUFFICIENT

Statement 2) X + Y = 5b+2
i.e. X+Y = 7 or 12 or 17 and so on...

Case 1: X+Y = 7 i.e. X = 5, Y = 2 then X^4 + Y^4 = 5^4 + 2^4 = Unit digit (5+6) = Unit digit 1
i.e. Remainder when divided by 5 = 1

Case 2: X+Y = 7 i.e. X = 6, Y = 1 then X^4 + Y^4 = 6^4 + 1^4 = Unit digit (6+1) = Unit digit 7
i.e. Remainder when divided by 5 = 2
INCONSISTENT SOLUTIONS

INSUFFICIENT

Combining the two Statements
X - Y = 5a+1 AND X + Y = 5b+2
Adding the two equation
2X = 5(a + b) + 3 => [for X to be an Integer]
i.e. X = 4 or 9 or 14... and so on

let's say X = 5c - 1

and Y = 5b+1 - X = 5b+1 - [5c-1] = 5 (b-c)+(2)
i.e. Y = (Multiple of 5)+(2)

let's say X = 5d + 2

i.e. X^4 + Y^4 = (5c-1)^4 + (5d+2)^4 when divided by 5 will always give us the remainder (1+6) i.e. Remainder = 2

SUFFICIENT

Answer: Option C
Last edited by GMATinsight on Thu Aug 07, 2014 9:55 am, edited 2 times in total.
"GMATinsight"Bhoopendra Singh & Sushma Jha
Most Comprehensive and Affordable Video Course 2000+ CONCEPT Videos and Video Solutions
Whatsapp/Mobile: +91-9999687183 l [email protected]
Contact for One-on-One FREE ONLINE DEMO Class Call/e-mail
Most Efficient and affordable One-On-One Private tutoring fee - US$40-50 per hour
Top
User avatar
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 » Thu Aug 07, 2014 8:42 am
GMATinsight wrote: Combining the two Statements
X - Y = 5a+1 AND X + Y = 5b+1
Adding the two equation
2X = 5a + 5b + 2
]i.e. X = Multiple of 5 + 1

let's say X = 5c+1

and Y = 5b+1 - X = 5b+1 - (5c+1) = 5 (b-c)
]i.e. Y = Multiple of 5

i.e. X^4 + Y^4 = (5c+1)^4 + (5d)^4 when divided by 5 will always give us the remainder 1

SUFFICIENT

Answer: Option C
Bhoopendra,

The equation in red should be X+Y = 5b + 2.
This typo seems to have lead to an erroneous conclusion.
When x� + y� is divided by 5, the remainder is not 1 but 2.
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
Top
User avatar
Legendary Member
Posts: 1100
Joined: Sat May 10, 2014 11:34 pm
Location: New Delhi, India
Thanked: 205 times
Followed by:24 members

by GMATinsight » Thu Aug 07, 2014 9:54 am
GMATGuruNY wrote:
Bhoopendra,

The equation in red should be X+Y = 5b + 2.
This typo seems to have lead to an erroneous conclusion.
When x� + y� is divided by 5, the remainder is not 1 but 2.
Thank you Mitch for pointing out. I have made the required correction. :)
"GMATinsight"Bhoopendra Singh & Sushma Jha
Most Comprehensive and Affordable Video Course 2000+ CONCEPT Videos and Video Solutions
Whatsapp/Mobile: +91-9999687183 l [email protected]
Contact for One-on-One FREE ONLINE DEMO Class Call/e-mail
Most Efficient and affordable One-On-One Private tutoring fee - US$40-50 per hour
Top
Post new topic Post Reply
• Page 1 of 1