Challenge Level

$n$ | $n^2$ | $n+4$ | remainder |
---|---|---|---|

1 | 1 | 5 | 1 |

2 | 4 | 6 | 4 |

3 | 9 | 7 | 2 |

4 | 16 | 8 | 0 |

5 | 25 | 9 | 7 |

6 | 36 | 10 | 6 |

7 | 49 | 11 | 5 |

... | ... | ||

Not much pattern so far | |||

... | ... | ||

20 | 400 | 24 | 24$\times$15 = 240 + 120 = 360 24$\times$16 = 360 + 24 = 384 remainder is 16 |

100 | 10000 | 104 | 104$\times$100 = 10400 104$\times$96 = 10400 - 4016 = 10000 - 16 remainder is 16 |

101 | 10201 | 105 | 105$\times$100 = 10500 105$\times$97 = 10500 - 315 = 10200 - 16 remainder is 16 |

Remainder is 16 for the larger numbers tested

Had to multiply by $n-4$ for the larger values of $n$, try this algebraically: $(n-4)(n+4)=n^2-16$

So $n^2 = \underbrace{(n-4)(n+4)}_{\text{multiple of }n+4} + 16$

So the remainder is the same as the remainder when $16$ is divided by $n+4$

This will be $16$ if $n+4\gt16$ so if $n\gt12$.

Check values for $n=8$ to $n=12$ ($n$ up to $7$ shown above)

$n$ | $n+4$ | remainder when $16$ divided by $n+4$ |
---|---|---|

8 | 12 | 4 |

9 | 13 | 3 |

10 | 14 | 2 |

11 | 15 | 1 |

12 | 16 | 0 |

This problem is taken from the UKMT Mathematical Challenges.

You can find more short problems, arranged by curriculum topic, in our short problems collection.