What is the remainde

[jp.jprasanna]

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

Source : gmatclub.com

OA : B

I have no idea how to apply algebra for this problem.
Testing nos took very long time >4mins and I wasn't even sure about the answer after testing nos.

Any suggestions?
Cheers

[kapilreadyforgmat]

I think C should be the answer, as there is following possibility for X, Y considering St 1 correct:
1. X=5N+1 & Y=5M+0 => reminder for X^4+Y^4 => 1
2. X=5N+4 & Y=5M+3 => reminder for X^4+Y^4 => 2
3. X=5N+5 & Y=5M+4 => reminder for X^4+Y^4 => 1
considering St 2 correct:
1. X=5N+2 & Y=5M+0 => reminder for X^4+Y^4 => 1
2. X=5N+4 & Y=5M+3 => reminder for X^4+Y^4 => 2

X=5N+4
Y=5M+3
Reminder will be 2 and only possible when St.1 and 2 both are considered.

Thanks
Kapil

[tim]

C looks correct to me unless i'm missing something. thanks for the algebraic approach; picking numbers should work out fine and within a reasonable time frame too if you jump right into the task and keep your focus..

[jp.jprasanna]

I think C should be the answer, as there is following possibility for X, Y considering St 1 correct:
1. X=5N+1 & Y=5M+0 => reminder for X^4+Y^4 => 1
2. X=5N+4 & Y=5M+3 => reminder for X^4+Y^4 => 2
3. X=5N+5 & Y=5M+4 => reminder for X^4+Y^4 => 1
considering St 2 correct:
1. X=5N+2 & Y=5M+0 => reminder for X^4+Y^4 => 1
2. X=5N+4 & Y=5M+3 => reminder for X^4+Y^4 => 2

X=5N+4
Y=5M+3
Reminder will be 2 and only possible when St.1 and 2 both are considered.

Thanks
Kapil

thanks Kapil and tim.... yes the answer should be C..

Kapil - can you please let me know how you arrived at the below equations for x and y using statement 1 and 2.

considering St 1 correct:
1. X=5N+1 & Y=5M+0 => reminder for X^4+Y^4 => 1
2. X=5N+4 & Y=5M+3 => reminder for X^4+Y^4 => 2
3. X=5N+5 & Y=5M+4 => reminder for X^4+Y^4 => 1
considering St 2 correct:
1. X=5N+2 & Y=5M+0 => reminder for X^4+Y^4 => 1
2. X=5N+4 & Y=5M+3 => reminder for X^4+Y^4 => 2

From statement 1 ... I can says X-Y = 5N+1 and from statement 2 X+Y = 5N + 2

So now we have 2 non-linear equation for x-y and x+y which we can use to solve for x and y. once we have the individual values of x and y we can substitute them back to the question to get the reminder. OK?

But I would like to know your approach as well. thanks.

[kapilreadyforgmat]

St 1: Reminder will be 1, if X-Y is divided by 5
=>
X=5N+something1
Y=5M+something2

We want
sthg1 - sthg2 = 1
1 - 0
2 - 1 (difference is divisible by 5)
3 - 2 (difference is divisible by 5)
4 - 3
5 - 4
After that it will start repeating i.e. 6= 5+1 etc.

In the same way St 2: we want sthg1+sthg2 = 2 or 7 (=>5+2)

sthg1 + sthg2 = 2 or 7
1 + 1
2 + 0
3 + 4
4 + 3

Prasanna: Your approach is infact better than mine. :-)

Thanks
Kapil

[tim]

thanks; let us know if there are still any questions about this one..