Name: Date:

Question 1: Prove that the product of 2 consecutive even numbers is always a multiple of 4 |

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

Question 3: Prove the identity (n+5)^2 -(9n+115) ≡ (n+10)(n-9) |

Question 4: Prove that (7n+9)^2 -(7n-9)^2 is always a multiple of 9 |

© GCSEMathsWorksheets.com

Show Answers
Hide Answers

© GCSEMathsWorksheets.com

# Answers

Question 1: 2n(2n+2) =4n^2+4n =4(n^2+n) Therefore the product of 2 consecutive even numbers is always a multiple of 4 |

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: (n+5)^2 -(9n+115) =(n^2+10n +25)-(9n+115) =n^2+n-90 =(n+10)(n-9) |

Question 4: (7n+9)^2 -(7n-9)^2 =(49n^2+126n +81)-(49n^2-126n +81) =252n=9 \times 28n Therefore (7n+9)^2 -(7n-9)^2 is a multiple of 9 |