  Given an $r$-colouring of the interval $\{2, \dotsc, N \}$, what is the
minimum number of monochromatic solutions of the equation $xy = z$? For $r=2$,
we show that there are always asymptotically at least $(1/2\sqrt{2}) N^{1/2}
\log N$ monochromatic solutions, and that the leading constant is sharp. We
also establish a stability version of this result. For general $r$, we show
that there are at least $C_r N^{1/S(r-1)}$ monochromatic solutions, where
$S(r)$ is the Schur number for $r$ colours and $C_r$ is a constant. This bound
is sharp up to logarithmic factors when $r \leq 4$. We also obtain results for
more general multiplicative equations of the form $x_1^{a_1} \dotsb x_k^{a_k} =
y$, where $a_1, \dotsc, a_k$ are positive integers, at least one of which
equals $1$. Our corresponding upper bounds are given in terms of certain
`interval Rado numbers' for additive equations. We pose a number of open
problems concerning these numbers.
