Show, for any integer n, if n is a multiple of 5, then n must be...

 

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Show, for any integer n, if n² is a multiple of 5, then n must be a multiple of 5 Use a combination of proof by contraposition and proof by cases. Start by writing the contrapositive of the statement to be proved. Then prove that statement using four cases: n is one more than a multiple of 5, two more than a multiple of 5, three more than a multiple of 5 or four more than a multiple of 5. In each of the four cases, establish that n² is of the form multiple of 5 plus something where something is either 1 or 4. Of course, either of those outcomes mean n² can't be an exact multiple of 5. Show, for any integer n, if n² is a multiple of 5, then n must be a multiple of 5 Use a combination of proof by contraposition and proof by cases. Start by writing the contrapositive of the statement to be proved. Then prove that statement using four cases: n is one more than a multiple of 5, two more than a multiple of 5, three more than a multiple of 5 or four more than a multiple of 5. In each of the four cases, establish that n² is of the form multiple of 5 plus something where something is either 1 or 4. Of course, either of those outcomes mean n² can't be an exact multiple of 5.

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