Prove the following For all integers ab and c if ab and ac t

Prove the following:

For all integers a,b, and c, if a|b and a|c, then a|(9b-4c).

Solution

For all integers n, 4(n² + n + 1) 3n² is a perfect square.

Proof:

Let n is any [particular but arbitrarily chosen] integer. [We must show that (4(n² + n + 1) 3n²) is a perfect square.] Then, we have

                                            4(n² + n + 1) 3n² = 4n² + 4n + 4 3n²

                                                                        = n² + 4n + 4

                                                                        = (n + 2)²

But is a perfect square [because (n+2) is an integer (being a sum of n and 2).] Hence, (4(n² + n + 1) 3n²) is an integer