I'm going through the following heat equation problem, and I understand how the solution is constructed except for one algebraic re-arrangement that I can't make sense of (and its probably really obvious) which is right at the start of the solution. I was hoping someone could enlighten me.

Solve $\displaystyle u_t = ku_{xx}$ on $\displaystyle 0 \leq x \leq L$ with IC's $\displaystyle u(x,0) = f(x)$ and BC's $\displaystyle u_x(0,t) = u_x(L,t) = 0$.

The example says:

Let $\displaystyle u = XT$.

Confused from here:
Sub in $\displaystyle XT' = kX''$

$\displaystyle \frac{X''}{X} = \frac{T'}{kT} = - \lambda$

Then $\displaystyle X'' + \lambda X = 0$.

*To the line above*
From here I have no problems understanding and solving this problem.

This is probably embarrassingly simple, but I keep getting

$\displaystyle \frac{X''}{X} = \frac{T'}{k}$

and why is this equal to $\displaystyle -\lambda$