Question

5. Let n be an integer. Do the following: (a) Prove that at least one of the following three statements is true: • There exists k ? Z such that $n^2 = 5k$. • There exists k ? Z such that $n^2 = 5k + 1$. • There exists k ? Z such that $n^2 = 5k + 4$. (b) Prove that exactly one of the three statements in part (a) is true. Hint. Since you already have the statement in (a), this is the same as showing that at most one of the three statements can be true; what do you think would go wrong if two of them were true? Also be aware that the letter k is being used in three independent statements in part (a), and that you shouldn't think of them all as \"the same k\". For instance, to say that the second and third statements are both true means that $n^2 = 5k_1 + 1$ for some $k_1 ? Z$ and $n^2 = 5k_2 + 4$ for some $k_2 ? Z$.

          5. Let n be an integer. Do the following:
(a) Prove that at least one of the following three statements is true:
• There exists k ? Z such that $n^2 = 5k$.
• There exists k ? Z such that $n^2 = 5k + 1$.
• There exists k ? Z such that $n^2 = 5k + 4$.
(b) Prove that exactly one of the three statements in part (a) is true.
Hint. Since you already have the statement in (a), this is the same as showing that at most one
of the three statements can be true; what do you think would go wrong if two of them were true?
Also be aware that the letter k is being used in three independent statements in part (a), and that
you shouldn't think of them all as \"the same k\". For instance, to say that the second and third
statements are both true means that $n^2 = 5k_1 + 1$ for some $k_1 ? Z$ and $n^2 = 5k_2 + 4$ for some
$k_2 ? Z$.

5. Let n be an integer. Do the following:
(a) Prove that at least one of the following three statements is true:
• There exists k ? Z such that n^2 = 5k.
• There exists k ? Z such that n^2 = 5k + 1.
• There exists k ? Z such that n^2 = 5k + 4.
(b) Prove that exactly one of the three statements in part (a) is true.
Hint. Since you already have the statement in (a), this is the same as showing that at most one
of the three statements can be true; what do you think would go wrong if two of them were true?
Also be aware that the letter k is being used in three independent statements in part (a), and that
you shouldn't think of them all as ẗhe same k.̈ For instance, to say that the second and third
statements are both true means that n^2 = 5k1 + 1 for some k1 ? Z and n^2 = 5k2 + 4 for some
k2 ? Z.

Added by Sean W.

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