Anaira Mitch wrote:Numbers a and b are positive integers. If a^4-b^4 is divided by 3, what is the remainder?
1) When a+b is divided by 3, the remainder is 0
2) When a^2+b^2 is divided by 3, the remainder is 2
a�-b� = (a²+b²)(a²-b²) = (a²+b²)(a+b)(a-b).
Statement 1:
Since a+b is divisible by 3, (a²+b²)(a+b)(a-b) is divisible by 3.
Thus, dividing a�-b� by 3 will yield a remainder of 0.
SUFFICIENT.
Statement 2:
In other words, a²+b² is 2 more than a multiple of 3:
a²+b² = 3k + 2, where k is a nonnegative integer.
Thus:
a²+b² = 2, 5, 8, 11, 14, 17, 20, 23...
Integer options for a and b such that a²+b² yields a value in the list above:
a=1 and b=1, with the result that a²+b² = 2 and
a-b = 0.
a=2 and b=1, with the result that a²+b² = 5 and
a+b = 3.
a=2 and b=2, with the result that a²+b² = 8 and
a-b = 0.
a=4 and b=1, with the result that a²+b² = 17 and
a-b= 3.
a=4 and b=2, with the result that a²+b² = 20 and
a+b = 6.
As the values in blue illustrate, every case yields a factor for a�-b� that is divisible by 3.
Thus -- in every case -- dividing a�-b� by 3 will yield a remainder of 0.
SUFFICIENT.
The correct answer is
D.
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