Problem (2), 10 points Let n be an integer. Prove that if 3 does not divide n, then 3(2n2 5)

Question

Question

Answer 1

Answer #1

Since n is not divisible by 3, n can be represented as 3m + 1 or 3m + 2.

Case 1: n = 3m + 1

2n² - 5 = 2(3m+1)²- 5 = 2(9m² + 6m + 1) - 5 = 18m² + 12m - 3 = 3(6m² + 4m - 1)

This proves that 3 | 2n² - 5

Case 2: n = 3m + 1

2n² - 5 = 2(3m+2)²- 5 = 2(9m² + 12m + 4) - 5 = 18m² + 24m + 3 = 3(6m² + 4m + 1)

This proves that 3 | 2n² - 5

Answer 2

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