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

## 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:

 $\ds 2184$ $=$ $\ds 2^3 \times 3 \times 7 \times 13$ $\ds 2185$ $=$ $\ds 5 \times 19 \times 23$ $\ds 2186$ $=$ $\ds 2 \times 1093$ $\ds 2187$ $=$ $\ds 3^7$ $\ds 2188$ $=$ $\ds 2^2 \times 547$ $\ds 2189$ $=$ $\ds 11 \times 199$ $\ds 2190$ $=$ $\ds 2 \times 3 \times 5 \times 73$ $\ds 2191$ $=$ $\ds 7 \times 313$ $\ds 2192$ $=$ $\ds 2^4 \times 137$ $\ds 2193$ $=$ $\ds 3 \times 17 \times 43$ $\ds 2194$ $=$ $\ds 2 \times 1097$ $\ds 2195$ $=$ $\ds 5 \times 439$ $\ds 2196$ $=$ $\ds 2^2 \times 3^2 \times 61$ $\ds 2197$ $=$ $\ds 13^3$ $\ds 2198$ $=$ $\ds 2 \times 7 \times 157$ $\ds 2199$ $=$ $\ds 3 \times 733$ $\ds 2200$ $=$ $\ds 2^3 \times 5^2 \times 11$

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:

 $\ds 2183$ $=$ $\ds 37 \times 59$ $\ds 2201$ $=$ $\ds 31 \times 71$

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

$\blacksquare$