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

This question has been treated in another thread, but I simply cannot filter

out the posts for reference.

The approach used in one of the post:

Each statement alone is not SUFFICIENT.

Let X - Y = 5A + 1

X + Y = 5B + 2

2X = 5(A+B) + 3

2Y = 5(B-A) + 1

16 (X4 + Y4) = {5(A+B)+3}4 + {5(B-A)+1}4

= (5P+3)4 + (5Q+1)4

= 81 + 1 + 5R

= 82 + 5R

X4 + Y4 = 5 + (5R+2)/16

Since L.H.S is multiple of 16, R.H.S. should also be multiple of 16(since R.H.S cannot be a fraction.).

Lowest value of R for which R.H.S. is an integer is R=6

X4 + Y4 = 82

Remainder of (X4 + Y4)/5 = 2Hence C

Could anyone assist to especially with the highlighted part. Thanks

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