What is the remainder when X^4 + Y^4 divide by 5
A) X-Y divided by 5 gives remainder 1
B) X+Y divided by 5 gives remainder 2
My answer is C and remainder is 2
We can easily found pair of number for which A and B are not self sufficient
Combining these,
(X+Y+X-Y) mod 5 = 3 mod 5
2X mod 5 = 3 mod 5
this gives X mod 5 = 4 as it can be satisfied for X = 4, 9 ....
Similarly, (X+Y-X+Y) mod 5 = 1 mod 5
i.e 2Y mod 5 = 1 mod 5 gives Y mod 5 = 3 as it is satisfied for Y = 3, 8...
so now X mod 5 = 4 and Y mod 5 = 3
i.e X^4 mod 5 = 4^4 mod 5 = 6 mod 5 = 1mod 5
Y^4 mod 5 = 1 mod 5
so X^4 + Y^4 mod 5 = (1+1) mod 5 = 2 mod 5
hence remainder 2.
Please let me know if reasoning and answer is correct.