Data Sufficiency

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

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.

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.

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.