Starting with three numbers $a, b, c$, at each step do one of the three operations:
- change $a$ to $2(b + c) - a$;
- change $b$ to $2(c + a) - b$;
- change $c$ to $2(a + b) - c$;
Define $f(a, b, c)$ to be the minimum number of steps required for one number to become zero. If this is not possible then $f(a, b, c)=0$.
For example, $f(6,10,35)=3$: $$(6,10,35) \to (6,10,-3) \to (8,10,-3) \to (8,0,-3).$$ However, $f(6,10,36)=0$ as no series of operations leads to a zero number.
Also define $F(a, b)=\sum_{c=1}^\infty f(a,b,c)$. You are given $F(6,10)=17$ and $F(36,100)=179$.
Find $\displaystyle\sum_{k=1}^{18}F(6^k,10^k)$.
Solution
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