Name: Date:

Question 1: Prove that the difference of the squares of 2 consecutive even numbers is always divisible by 4 |

Question 2: Prove the identity (n-5)^2 -(6n-35) ≡ (n-10)(n-6) |

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

Question 4: Prove that (8n+5)^2 -(8n-5)^2 is always a multiple of 5 |

© GCSEMathsWorksheets.com

Show Answers
Hide Answers

© GCSEMathsWorksheets.com

# Answers

Question 1: (2n+2)^2-(2n)^2 =4n^2+8n+4-4n^2 =8n+4=4(2n+1) therefore the difference of the squares of 2 consecutive even numbers is always divisible by 4 |

Question 2: (n-5)^2 -(6n-35) =(n^2-10n +25)-(6n-35) =n^2-16n+60 =(n-10)(n-6) |

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: (8n+5)^2 -(8n-5)^2 =(64n^2+80n +25)-(64n^2-80n +25) =160n=5 \times 32n Therefore (8n+5)^2 -(8n-5)^2 is a multiple of 5 |