Factor the expression:
n4−1=(n2−1)(n2+1)=(n−1)(n+1)(n2+1)==(n−1)(n+1)((n2−4)+5)=(n−1)(n+1)((n−2)(n+2)+5)==(n−1)(n+1)(n−2)(n+2)+5(n−1)(n+1) The second term of the expression is divisible by 5.
By the division algorithm, any n∈Z can be expressed uniquely in the form:
n=5q+r, where q∈Z and r=0,1,2,3,4 .
If n=5q, then (n−1)(n+1)(n−2)(n+2)=(5q−1)(5q+1)(5q−2)(5q+2) is not divisible by 5.
If n=5q+1, then the first term is 5q(5q+2)(5q−1)(5q+3), it is divisible by 5.
If n=5q+2, then (5q+1)(5q+3)⋅5q⋅(5q+4) is divisible by 5.
If n=5q+3, then (5q+2)(5q+4)(5q+1)(5q+5)=5(5q+2)(5q+4)(5q+1)(q+1) is divisible by 5.
If n=5q+4, then the first term is (5q+3)(5q+5)(5q+2)(5q+6)=5(5q+3)(q+1)(5q+2)(5q+6), it is divisible by 5.
Thus, n4−1 is divisible by 5 when n is not divisible by 5 (n=/5k).
Answer: all positive integers except multiples of 5.
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