# 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:

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

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.

$\blacksquare$