Since there are four questions, and each answer can be yes or no, there are sixteen possible combinations of answers (Think of these like binary bits of a four bit number).

Here are are all sixteen of these answer combinations enumerated and, on the right, which ‘secrets’ would trigger these answers.

Divisible | —— |

7 | 5 | 3 | 2 | Solutions | —— |

N | N | N | N | 1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 | |

N | N | N | Y | 2, 4, 8, 16, 22, 26, 32, 34, 38, 44, 46, 52, 58, 62, 64, 68, 74, 76, 82, 86, 88, 92, 94 | |

N | N | Y | N | 3, 9, 27, 33, 39, 51, 57, 69, 81, 87, 93, 99 | |

N | N | Y | Y | 6, 12, 18, 24, 36, 48, 54, 66, 72, 78, 96 | |

N | Y | N | N | 5, 25, 55, 65, 85, 95 | |

N | Y | N | Y | 10, 20, 40, 50, 80, 100 | |

N | Y | Y | N | 15, 45, 75 | |

N | Y | Y | Y | 30, 60, 90 | |

Y | N | N | N | 7, 49, 77, 91 | |

Y | N | N | Y | 14, 28, 56, 98 | |

Y | N | Y | N | 21, 63 | |

Y | N | Y | Y | 42, 84 | |

Y | Y | N | N | 35 | |

Y | Y | N | Y | 70 | |

Remember that Lucy said that the answers to the questions uniquely determine her number? This means we are looking for rows that have just one solution. As it happens, there are only two; the last two rows. The secret is therefore 35, or 70.If you look closely, you will see that both of these solutions have the same answers to the questions “Divisible by seven?”, “Divisible by five?” and “Divisible by three?”. Because of this, we don’t need to know the answers to these questions!The question we need to know the answer to is “Divisible by two?”As we heard this answer is “Yes”, then Lucy’s secret number is 70. |

If you look closely, you will see that both of these solutions have the same answers to the questions *“Divisible by seven?”, “Divisible by five?”* and *“Divisible by three?”*. Because of this, we don’t need to know the answers to these questions!

As we heard this answer is *“Yes”*, then Lucy’s secret number is 70.

*Lucy’s Secret Number is 70.*