The sixth product

The six products are $ab, ac, ad, bc, bd$ and $cd$ . By pairing $ab$ and $cd$ , the product is $abcd$ . Likewise for pairing $ac$ and $bd$ as well as $ad$ and $bc$ . From the 5 products $2, 3, 5, 6, 9$ , we have $2 \times 9 =3\times 6 =18$ . This suggests that the sixth product is $\frac{18}{5}=3.6$ . So, the integer closest to $3.6$ is $\large \textcolor{#D61F06} 4$ .

Interestingly, there are more than one possible values for $(a,b,c,d)$ . For example: $(a, b, c, d) = \left( \sqrt{\frac{5}{3}} , \sqrt{\frac{12}{5}}, \frac{3}{2}\sqrt{\frac{12}{5}}, 3\sqrt{\frac{5}{3} } \right) , \left( \sqrt{\frac{6}{5}} , \sqrt{\frac{10}{3}}, \frac{3}{2}\sqrt{\frac{10}{3}}, 3\sqrt{\frac{6}{5} } \right)$ .

As all the numbers are equivalent, let the unknown pair is $cd$ .

Now, $\frac{ad}{ac}=\frac{bd}{bc}$ .

There is one such combination ( $(6,2);(3,3)$ ).

So, take $ad=6, ac=2, bd=9, bc=3$ . This doesn't affect the generality.

Similarly, $\frac{ab}{bc}=\frac{5}{3}=\frac{ad}{cd} \rightarrow cd=\frac{6}{\frac{3}{5}}=\frac{18}{5}$ .

So, the nearest integer to this is $4$ .