Find the GCD of $2^{60}-1$ and $2^{50}-1$

Question

My teacher said we can solve this using Euclid's division lemma but I have no clue how. I factored the numbers into $$2^{60}-1 = (2^{10}-1)(1+2^{10}+2^{20}+...+2^{50})$$ and $$2^{50}-1 = (2^{10}-1)(1+2^{10}+2^{20}+...+2^{40})$$ So I got the gcd as $2^{10}-1$. But I don't know if there can be some common factor in $(2^{10}-1)(1+2^10+2^{20}+...+2^{40})$ and $(2^{10}-1)(1+2^{10}+2^{20}+...+2^{50})$.