#question in proving a statement

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sharp gardenBOT
visual apex
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Can someone help me please :/

warped idol
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You proved the opposite implication, i.e. if 3|n then 3|n^2. To prove the right implication, start with 3|n^2.

Suppose that 3 divides n^2

Case a: 3 divides n

Case b: Suppose there is a remainder of 1 when 3 divides
n. Then n = 3k + 1 for some integer k.

Then n^2 = (3k +1)^2 = 9k^2 + 6k + 1. Since 3 divides 9k 2 + 6k, therefore when 3 divides n^2 there is a remainder of 1. So case b cannot occur when 3 divides n^2

Case c: Suppose there is a remainder of 2 when 3 divides
n. Then n = 3k + 2 for some integer k.

Then n^2 = (3k + 2)^2 = 9k^2 + 12k + 4. Since 3 divides 9k^2 + 12k + 3, when 3 divides n^2 there is a remainder of 1 again. So case c
cannot occur when 3 divides n^2.

That leaves only case a in which 3 divides n.
Therefore, if 3 divides n^2, then 3 also divides n.

visual apex
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Ye nvm i got it

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Thank u anyways tho

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:))

drowsy sparrow
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.solved