# 17 Consecutive Integers each with Common Factor with Product of other 16

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## Theorem

The $17$ consecutive integers from $2184$ to $2200$ have the property that each one is not coprime with the product of the other $16$.

## Proof

We obtain the prime decomposition of all $17$ of these integers:

\(\displaystyle 2184\) | \(=\) | \(\displaystyle 2^3 \times 3 \times 7 \times 13\) | |||||||||||

\(\displaystyle 2185\) | \(=\) | \(\displaystyle 5 \times 19 \times 23\) | |||||||||||

\(\displaystyle 2186\) | \(=\) | \(\displaystyle 2 \times 1093\) | |||||||||||

\(\displaystyle 2187\) | \(=\) | \(\displaystyle 3^7\) | |||||||||||

\(\displaystyle 2188\) | \(=\) | \(\displaystyle 2^2 \times 547\) | |||||||||||

\(\displaystyle 2189\) | \(=\) | \(\displaystyle 11 \times 199\) | |||||||||||

\(\displaystyle 2190\) | \(=\) | \(\displaystyle 2 \times 3 \times 5 \times 73\) | |||||||||||

\(\displaystyle 2191\) | \(=\) | \(\displaystyle 7 \times 313\) | |||||||||||

\(\displaystyle 2192\) | \(=\) | \(\displaystyle 2^4 \times 137\) | |||||||||||

\(\displaystyle 2193\) | \(=\) | \(\displaystyle 3 \times 17 \times 43\) | |||||||||||

\(\displaystyle 2194\) | \(=\) | \(\displaystyle 2 \times 1097\) | |||||||||||

\(\displaystyle 2195\) | \(=\) | \(\displaystyle 5 \times 439\) | |||||||||||

\(\displaystyle 2196\) | \(=\) | \(\displaystyle 2^2 \times 3^2 \times 61\) | |||||||||||

\(\displaystyle 2197\) | \(=\) | \(\displaystyle 13^3\) | |||||||||||

\(\displaystyle 2198\) | \(=\) | \(\displaystyle 2 \times 7 \times 157\) | |||||||||||

\(\displaystyle 2199\) | \(=\) | \(\displaystyle 3 \times 733\) | |||||||||||

\(\displaystyle 2200\) | \(=\) | \(\displaystyle 2^3 \times 5^2 \times 11\) |

It can be seen by inspection that each of the integers in this sequence shares at least one prime factor with at least one other.

It is then worth noting that:

\(\displaystyle 2183\) | \(=\) | \(\displaystyle 37 \times 59\) | |||||||||||

\(\displaystyle 2201\) | \(=\) | \(\displaystyle 31 \times 71\) |

and it can be seen that the sequence can be extended neither upwards nor downwards.

$\blacksquare$

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2185$