Using a direct proof how do I prove that if n is divisible b

Using a direct proof, how do I prove that if n is divisible by 3, then the number n^2 +3n is divisible by 9?

2. Prove using a direct proof, that n is divisible by 3, then the number n 3n is divisible if by 9 Assume n is odd No 30 on t on t n Since Assam ng antecendent s true conclu is False since n 13n is ble by a Page 2 of 6

Solution

Dear Student

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To prove that if n is divisible by 3 then n^2 + 3n is divisible by 9

Proof

If n is divisible by 3 => n can be assumed to be equal to 3k where is an integer.

=> n = 3k

Now substituting n in given polynomial n^2 + 3n

= (3k)^2 + 3 (3k)

= 9k^2 + 9k

= 9 (k^2 + k)

Now since k is an integer, therefore k^2 + k is also an integer

Hence n is a multiple of 9 as clearly proved above.