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Solution:

Why are the prime numbers clustered around multiples of 6?

Let's consider what happens when we divide an integer by 6. There are 6 possible remainders, including the remainder of 0, which we get when the number is a multiple of 6:

Remainder when we divide by 6 | Can the number be prime? |

0 | Must be a multiple of 2 and 3. |

1 | May be prime, such as 13. But not always, such as 25. |

2 | Must be a multiple of 2. |

3 | Must be a multiple of 3. |

4 | Must be a multiple of 2. |

5 | May be prime, such as 29. But not always, such as 35. |

Looking at the chart, we can see that numbers leaving a remainder of 0, 2, or 4 must be a multiple of 2, and numbers leaving a remainder of 0 or 3 must be a multiple of 3. Other than 2 and 3 themselves, these numbers are all composite. Therefore *all* primes greater than 3 must have a remainder of 1 or 5 when we divide by 6. In other words, they must be one more or one less than a multiple of 6.

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