# Divisibility Rules

## Divisibility

If **a** and **m** are two integers, than **a **is a divisor of** m**, or **m** is divisible by **a** when:

$$a|m\iff \exists n:n\in \mathrm{\mathbb{N}}\wedge n\xb7a=m$$

## Divisibility Rules

$$\left(a|m\right)\wedge \left(a|n\right)\Rightarrow \left(a|m\xb7n\right)\wedge \left(a|\left(m+n\right)\right)\wedge \left(a\left|\right|m-n|\right)$$

$$\left(a|b\right)\wedge \left(b|c\right)\Rightarrow a|c$$

$$\left(p\in \mathrm{\mathbb{P}}\right)\wedge \left(p|m\xb7n\right)\Rightarrow \left(p|m\right)\vee \left(p|n\right)$$

## Divisibility rules for numbers

2|n |
last digit of n ∈{0,2,4,6,8} |

3|n |
sum of all digts is divisible by 3 |

4|n |
last 2 digits are divisible by 4 |

5|n |
last digit of n ∈{0,5} |

6|n |
n is divisible by both 2 and 3 |

8|n |
last three digits are divisible by 8 |

9|n |
sum of all digts is divisible by 9 |

10|n |
last digit of n is 0 |

25|n |
last two digits of n ∈{100,25,50,75} |

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