Answer
(a) The force constant is 2k.
(b) The force constant is 3k.
Work Step by Step
Suppose the original spring pulls with a force of $F$ when stretched a distance $x$.
$F = kx$
$k = \frac{F}{x}$
(a) If the spring is cut in half, each half of the spring pulls with a force of $F$ when stretched a distance of $\frac{x}{2}$
$F = k_2\frac{x}{2}$
$k_2 = \frac{2F}{x} = 2k$
The force constant is 2k.
(b) If the spring is cut into three segments, each segment of the spring pulls with a force of $F$ when stretched a distance of $\frac{x}{3}$
$F = k_3\frac{x}{3}$
$k_3 = \frac{3F}{x} = 3k$
The force constant is 3k.
