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Question 1: Prove that the difference of the squares of 2 consecutive numbers is always the sum of the 2 numbers |

Question 2: Prove the identity (n+3)^2 -(4n+33) ≡ (n+6)(n-4) |

Question 3: Prove that the sum of 3 consecutive numbers is always a multiple of 3 |

Question 4: Prove that (6n+7)^2 +(6n-7)^2 is always even |

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Question 1: (2n+1)^2-(2n)^2 =4n^2+4n+1-4n^2 =4n+1=(2n+1)+(2n) therefore the difference of the squares of 2 consecutive numbers is always the sum of the 2 numbers |

Question 2: (n+3)^2 -(4n+33) =(n^2+6n +9)-(4n+33) =n^2+2n-24 =(n+6)(n-4) |

Question 3: n+(n+1)+(n+2) =3n+3 =3(n+1) Therefore the sum of 3 consecutive numbers is always a multiple of 3 |

Question 4: (6n+7)^2 +(6n-7)^2 =(36n^2+84n +49)+(36n^2-84n +49) =72n^2+98=2(36n^2+49) (6n+7)^2 +(6n-7)^2 is a multiple of 2, therefore it is even |