#math trivia for #June16: #168 is 2^3 * 3^1 * 7^1. How many positive divisors does it have?

— Burt Kaliski Jr. (@modulomathy) June 16, 2012

The number of positive divisors of 168 is 16. Given the factorization of 168, we know that any positive divisor must have the form

2* ^{i}* * 3

** 7*

^{j}

^{k}where *i* is between 0 and 3, *j* is 0 or 1, and *k* is 0 or 1. All together, there are 4*2*2 = 16 possible combinations of *i*, *j* and *k*. (No two combinations produce the same divisor, or otherwise a divisor would have two different prime factorizations, which is not possible according to the Unique Factorization Theorem.)

The 16 divisors and their corresponding factorizations are listed in the following table for reference.

Divisor | Factorization |

1 | 2^{0} * 3^{0} * 7^{0} |

2 | 2^{1} * 3^{0} * 7^{0} |

3 | 2^{0} * 3^{1} * 7^{0} |

4 | 2^{2} * 3^{0} * 7^{0} |

6 | 2^{0} * 3^{1} * 7^{0} |

7 | 2^{0} * 3^{0} * 7^{1} |

8 | 2^{3} * 3^{0} * 7^{0} |

12 | 2^{2} * 3^{1} * 7^{0} |

14 | 2^{1} * 3^{0} * 7^{1} |

21 | 2^{0} * 3^{1} * 7^{1} |

24 | 2^{3} * 3^{1} * 7^{0} |

28 | 2^{2} * 3^{0} * 7^{1} |

42 | 2^{2} * 3^{1} * 7^{1} |

56 | 2^{3} * 3^{0} * 7^{1} |

84 | 2^{2} * 3^{1} * 7^{1} |

168 | 2^{3} * 3^{1} * 7^{1} |

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