GMAT Question of the Day - Daily to your Mailbox; hard ones only

It is currently 23 Sep 2018, 20:49

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

What is the remainder when x^4 + y^4 divided by 5?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Manager
Manager
User avatar
B
Status: GMAT Coach
Joined: 05 Nov 2012
Posts: 135
Location: Peru
GPA: 3.98
Re: What is the remainder when x^4 + y^4 divided by 5?  [#permalink]

Show Tags

New post 20 Oct 2016, 12:23
Sunchaser20 wrote:
Well, that solution on that forum says:
A) X-Y divided by 5 gives remainder 1, then X-Y=5a+1
B) X+Y divided by 5 gives remainder 2, then X+Y=5b+2

Sum up A) and B): X-Y+X+Y=5a+1+5b+2 -> 2X=5(a+b)+3: when 2X is divided by 5, the remainder is 3. Then, when X is divided by 5, the remainder is 4, as the only values that satisfy 2X=5(a+b)+3 are 4; 9; 14... etc.
Subtract A) from B): X+Y-X+Y=5b+2-5a-1 -> 2Y=5(b-a)+1: when 2X is divided by 5, the remainder is 1. Then, when Y is divided by 5, the remainder is 3, as the only values that satisfy 2Y=5(b-a)+1 are 3; 8; 13... etc.
Now, knowing that:
- X divided by 5 gives remainder 4, and
- Y divided by 5 gives remainder 3
we can calculate the remainder when \(X^4+Y^4\) is divided by 5, and the answer is C.

But I think, that the answer is E, because:
Sum up A) and B): X-Y+X+Y=5a+1+5b+2 -> 2X=5(a+b)+3, when 2X is divided by 5, the remainder is 3. Then, when X is divided by 5, the remainder is 4 or 1.5 (X is not integer), as the values that satisfy 2X=5(a+b)+3 are 4; 6.5; 9; 11.5; 14... etc.
Subtract A) from B): X+Y-X+Y=5b+2-5a-1 -> 2Y=5(b-a)+1, when 2X is divided by 5, the remainder is 1. Then, when Y is divided by 5, the remainder is 3 or 0.5 (Y is not integer), as the values that satisfy 2Y=5(b-a)+1 are 3; 5.5; 8; 10.5; 13... etc.
Then, when \(X^4+Y^4\) will result in decimal without "remainders" when divided by 5.

So, the answer is E unless we are given that X and Y are integers, then the answer will be C.


The problem does not need to specify that the variables are integers; if the question asks for a remainder, then there must be a remainder. The remainder is always an integer; if the answer is not an integer, then it is not a remainder; therefore it is not a valid answer.

Does this help?
_________________

Clipper Ledgard
GMAT Coach

Manager
Manager
User avatar
B
Status: GMAT Coach
Joined: 05 Nov 2012
Posts: 135
Location: Peru
GPA: 3.98
Re: What is the remainder when x^4 + y^4 divided by 5?  [#permalink]

Show Tags

New post 20 Oct 2016, 12:32
skpMatcha wrote:
Sure thanks for the explanation.

what threw me off is.. the remainder is 4 or 1.5(X is not an integer). I was expecting by your explanation , when (a+b is even ). So got confused but now realized that they are one and the same :)

I hope real GMAT doesnt ask such heavy qns :(

Quote:
Sum up A) and B): X-Y+X+Y=5a+1+5b+2 -> 2X=5(a+b)+3, when 2X is divided by 5, the remainder is 3. Then, when X is divided by 5, the remainder is 4 or 1.5 (X is not integer), as the values that satisfy 2X=5(a+b)+3 are 4; 6.5; 9; 11.5; 14... etc.

The problem does not need to specify that the variables are integers; if the question asks for a remainder, then there must be a remainder. The remainder is always an integer; if the answer is not an integer, then it is not a remainder; therefore it is not a valid answer.

Does this help?
_________________

Clipper Ledgard
GMAT Coach

Retired Moderator
User avatar
B
Joined: 05 Jul 2006
Posts: 1731
GMAT ToolKit User Premium Member
Re: What is the remainder when x^4 + y^4 divided by 5?  [#permalink]

Show Tags

New post 20 Oct 2016, 14:02
mdfrahim wrote:
What is the remainder when x^4 + y^4 divided by 5?

(1) x - y divided by 5 gives remainder 1
(2) x + y divided by 5 gives remainder 2

I got this from some other forum. Here is the link
http://www.manhattangmat.com/forums/x4- ... t7277.html
I was not able to undersand the solution at all. I am sure on this forum I will definitely be able to get an answer which I am looking for.


It is pure algebra ....

2(x^4+y^4) = (x^2+y^2)^2 + (x^2-y^2)^2 and 2(x^2+y^2) = (x+y)^2 + (x-y)^2

each alone is insuff
together

(x-y) = 5m+1, x+y = 5n+2

(x^2 - y^2)^2 = {(x-y)(x+y)]^2 = (5m+1)^2* (5n+2)^2 = (25m+10m+1)*(25n+10n+4) = ...... only 1*4 is not multiple of 5 and giver r= 4

(x+y)^2 + (x-y)^2 = 25m+10m+1+25n+10n+4 = a multiple of 5 thus (x^2+y^2)^2 is multiple of 5

thus 2(x^4+y^4) always gives a remainder of 4 when divided by 5 therefore x^4+y^4 gives a remainder of 2 when divided by 5
Intern
Intern
avatar
B
Joined: 01 Jun 2011
Posts: 20
Re: What is the remainder when x^4 + y^4 divided by 5?  [#permalink]

Show Tags

New post 07 Sep 2018, 01:25
The remainder upon devision by 5 can be found by seeing the units digit of a number. The remainder can be 0.1.2.3.4

I tried the units digit approach.

x-y div by 5 gives 1 as remainder.

find two integers who upon subtraction give 1 or 6 as units digit.

4-3 gives 1
6-5 gives 1 as remainder
9-3 6 as remainder etc

we can raise them to 4th power in each case and check the unites digit of the sum x4+y4

4,3 will give 1 as units digit and 6+1 =7 as units digit of sum giving 2 as remainder
6-5 will give 1 as remainder of subtraction and of the sum 1 as remainder upon div by 5

statement 1 is insufficient

Similarity stat 2

we need to sum two number whose units digit gives us 2 as units digit

9+3 gives 2
7+5 gives 2

when we raise these to 4th power we get 2 as remainder in the first case and 1 in the second case insufficient

now looking at both statements 9 and 3 as units digits satisfies both 9-3 =6 for first statement and 9+3 for second.

other combinations don't satisfy both statements.


Comments?
GMAT Club Bot
Re: What is the remainder when x^4 + y^4 divided by 5? &nbs [#permalink] 07 Sep 2018, 01:25

Go to page   Previous    1   2   [ 24 posts ] 

Display posts from previous: Sort by

What is the remainder when x^4 + y^4 divided by 5?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  

Events & Promotions

PREV
NEXT


Copyright

GMAT Club MBA Forum Home| About| Terms and Conditions and Privacy Policy| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.