Brilliant isn't a waste of time!

let the number be $10n+y\longrightarrow n,y\le 9$ we get $\begin{cases} n+y=6\\ ny-n=6\\ \dfrac{y}{n}+y=6 \longrightarrow ny+y=6n\\ \end{cases}$ adding the first 2 we get $\begin{cases} ny+y=12\\ ny+y=6n\\ n=2\\y=4 \end{cases}$ and $\therefore 10n+y=20+4=\boxed{24}$

Whee.

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def goal(x):
    s = str(x)
    a = int(s[0]) + int(s[1])
    if a != 6:
        return False
    b = int(s[0]) * int(s[1]) - int(s[0])
    if b != 6:
        return False
    c = float(s[1])/int(s[0])+int(s[1])
    if c == 6:
        return True
for i in xrange(10,100):
    if goal(i):
        print "Answer:", i
        break