Answer to Question #97807 in Combinatorics | Number Theory for Anonymous

Question #97807
Find all positive integers n such that n^4 - 1 is divisible by 5
1
Expert's answer
2019-11-04T09:19:21-0500

Factor the expression:


n41=(n21)(n2+1)=(n1)(n+1)(n2+1)=n^4-1=(n^2-1)(n^2+1)=(n-1)(n+1)(n^2+1)==(n1)(n+1)((n24)+5)=(n1)(n+1)((n2)(n+2)+5)==(n-1)(n+1)((n^2-4)+5)=(n-1)(n+1)((n-2)(n+2)+5)==(n1)(n+1)(n2)(n+2)+5(n1)(n+1)=(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 nZn\in Z can be expressed uniquely in the form:


n=5q+r,n=5q+r,

where qZq\in Z and r=0,1,2,3,4r={0,1,2,3,4} .

If n=5qn=5q, then (n1)(n+1)(n2)(n+2)=(5q1)(5q+1)(5q2)(5q+2)(n-1)(n+1)(n-2)(n+2)=(5q-1)(5q+1)(5q-2)(5q+2) is not divisible by 5.

If n=5q+1n=5q+1, then the first term is 5q(5q+2)(5q1)(5q+3)5q(5q+2)(5q-1)(5q+3), it is divisible by 5.

If n=5q+2n=5q+2, then (5q+1)(5q+3)5q(5q+4)(5q+1)(5q+3)\cdot 5q\cdot (5q+4) is divisible by 5.

If n=5q+3n=5q+3, then (5q+2)(5q+4)(5q+1)(5q+5)=5(5q+2)(5q+4)(5q+1)(q+1)(5q+2)(5q+4)(5q+1)(5q+5)=5(5q+2)(5q+4)(5q+1)(q+1) is divisible by 5.

If n=5q+4n=5q+4, then the first term is (5q+3)(5q+5)(5q+2)(5q+6)=5(5q+3)(q+1)(5q+2)(5q+6)(5q+3)(5q+5)(5q+2)(5q+6)=5(5q+3)(q+1)(5q+2)(5q+6), it is divisible by 5.

Thus, n41n^4-1 is divisible by 5 when n is not divisible by 5 (n=/5k)(n{=}\mathllap{/\,}5k).

Answer: all positive integers except multiples of 5.


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