If a and b are integers such that a - b = 10, find the minimum value of a*b.

Guest Jun 16, 2021

#2**+1 **

The term $a*b$ is nonnegative whenever both $a$ and $b$ have the same sign. It is negative whenever $a$ is positive and $b$ is negative. Since $a$, $b$ are integers with $a = b+10$, these are all the possible ways that $ab$ can be negative:

\begin{eqnarray*}

a = 1, b = -9, ab = -9 \\

a = 2, b = -8, ab = -16 \\

a = 3, b = -7, ab = -21 \\

a = 4, b = -6, ab = -24 \\

a = 5, b = -5, ab = -25 \\

a = 6, b = -4, ab = -24 \\

a = 7, b = -3, ab = -21 \\

a = 8, b = -2, ab = -16 \\

a = 9, b = -1, ab = -9

\end{eqnarray*}

So the minimum value of $ab$ is $-25$.

Bginner Jun 16, 2021