...integers n...? k is an integer such that n3 + k is not divisible by 4 for all integers n.
What are the possible values of k?
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...integers n...? k is an integer such that n3 + k is not divisible by 4 for all integers n.
What are the possible values of k?
Need to find k such that n^3 + k is not divisible by 4 for all n.
Need to find k such that n^3 + k =/= 0 (mod 4)
There are four incongruent possibilities for k: 0, 1, 2, and 3.
If k = 0, then the question is does n^3 =/= 0 for all n? No because n = 0 gives 0^3 = 0 (mod 4)
If k = 1, then the question is does n^3 + 1 =/= 0 for all n? No because n = 3 gives 3^3 + 1 = 28 = 4(7) = 0 (mod 4)
If k = 2, then the question is does n^3 + 2 =/= 0 for all n?
Yes if n = 0, then n^3 + k = 2 =/= 0 (mod 4)
and if n = 1, then n^3 + k = 1^3 + 2 = 3 =/= 0 (mod 4)
and if n = 2, then n^3 + k = 2^3 + 2 = 10 =/= 0 (mod 4)
and if n = 3, then n^3 + k = 3^3 + 2 = 29 =/= 0 (mod 4)
If k = 3, then the question is does n^3 + 3 =/= 0 for all n? No because n = 1 gives 1^3 + 3 = 4 = 0 (mod 4)
Thus, the only solution is k = 2.
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