Name: Date:

Question 1: Prove the identity (n-4)^2 -(7n-40) ≡ (n-8)(n-7) |

Question 2: Prove that the difference of the squares of 2 consecutive odd numbers is always divisible by 8 |

Question 3: Prove that (8n+7)^2 -(8n-7)^2 is always a multiple of 8 |

Question 4: Prove that the product of 2 consecutive odd numbers is always odd |

© GCSEMathsWorksheets.com

Show Answers
Hide Answers

© GCSEMathsWorksheets.com

# Answers

Question 1: (n-4)^2 -(7n-40) =(n^2-8n +16)-(7n-40) =n^2-15n+56 =(n-8)(n-7) |

Question 2: (2n+3)^2-(2n+1)^2 =4n^2+12n+9-(4n^2+4n+1) =8n+8=8(n+1) therefore the difference of the squares of 2 consecutive odd numbers is always divisible by 8 |

Question 3: (8n+7)^2 -(8n-7)^2 =(64n^2+112n +49)-(64n^2-112n +49) =224n=8 \times 28n Therefore (8n+7)^2 -(8n-7)^2 is a multiple of 8 |

Question 4: (2n+1)(2n+3) =4n^2+8n+3 =2(2n^2+4n)+3 Any multiple of 2 is always even therefore 2(2n^2+4n) is even. Even+Odd=Odd therefore the product of 2 consecutive odd numbers is always odd |