# What is the Sixth Number?

I'm thinking of 4 real numbers, $a,b,c$ and $d$ (they need not be integers, or even rational). There are 6 products you can make by multiplying pairs of these numbers, namely, $ab,ac,ad,bc,bd$ and $cd$. I'll tell you what five of those products are, in no particular order: $$2,3,4,5 \text{ and }6$$ What is the sixth product?

Small hint:

You can do this without needing to figure out what $a,b,c$ and $d$ are.

This is a fun puzzle I found on the xkcd puzzle wiki.

• Is it 1 or 7??? – ʇolɐǝz ǝɥʇ qoq Mar 28 '15 at 0:23
• Do a,b,c,d have to be integers? – Rand al'Thor Mar 28 '15 at 0:24
• They need not be, question has been clarified, though you solved it anyway :) – Mike Earnest Mar 28 '15 at 0:43

The six numbers can be split up into pairs such that the product of the two numbers in each pair is the same across all three pairs: $(ab,cd),(ac,bd),(ad,bc)$. The only two pairs of numbers among $2,3,4,5,6$ which have the same product are $(3,4),(2,6)$. So the sixth number must be
$\frac{12}{5}$.