Malaysia 15

Find all positive integers $a, b, c$ such that $2^{a} 3^{b}=7^{c}-1$.
If $c>1,7^{c}-1$ has prime divisors that $7-1$ does not have according to Zsigmondy's theorem, thus other than 2 and 3. Therefore, $c=1$, and consequently $a=b=1$, which gives a unique solution.