How does one solve these type of some without using calculator?
2026-03-30 09:17:04.1774862224
Computing $\gcd of\left(\left(5\displaystyle\prod_{i=1}^{7}63^i\right)^{10},~135^{135}\right)$
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\begin{align*} \gcd((5\prod_{i = 1}^7 63^i)^{10},135^{135}) & = \gcd((5^{10}(63^{1+2+\dots+7})^{10},135^{135}) \\ & = \gcd(5^{10}63^{28\times 10},(5\times 27)^{5\times 27}) \\ & = \gcd(5^{10}(3\times 3\times 7)^{280},5^{5\times 3^3}3^{3\times 5\times 3^3}) \\ & = 5^{10}\times 3^{135\times 3} \end{align*}