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
Reminders
This topic has expert replies
-
- Master | Next Rank: 500 Posts
- Posts: 468
- Joined: Mon Jul 25, 2011 10:20 pm
- Thanked: 29 times
- Followed by:4 members
- 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 integer m is divided by 5, the remainder depends on the UNITS digit of m.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
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
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
- GMATinsight
- Legendary Member
- Posts: 1100
- Joined: Sat May 10, 2014 11:34 pm
- Location: New Delhi, India
- Thanked: 205 times
- Followed by:24 members
Question : Remainder when X^4 + Y^4 is divided by 5 ?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
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
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
- 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
Bhoopendra,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
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
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
- GMATinsight
- Legendary Member
- Posts: 1100
- Joined: Sat May 10, 2014 11:34 pm
- Location: New Delhi, India
- Thanked: 205 times
- Followed by:24 members
Thank you Mitch for pointing out. I have made the required correction.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.
"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
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