Compute the greatest common divisor of $ \ 300⋅35⋅7^{5}\ $ and $ \ 33^{5} ⋅3⋅64$

Question

Question:

Compute the greatest common divisor of $ \ 300⋅35⋅7^{5}\ $ and $ \ 33^{5} ⋅3⋅64$ using Euclidean algorithm.

I am not even sure which of the two numbers is bigger. Is there any way to figure that out? I need to write it in the form $ \ a = bq + r$ to start.

Answer 1

To find which is larger without a calculator I would note that $\frac {33}7$ is about $5$, so the fifth power terms contribute five extra factors of $5$ on the right. $\frac {300}{64}$ will take care of one and $\frac {35}3$ will take care of less than two, so $33^5 \cdot 3 \cdot 64 \gt 300 \cdot 35 \cdot 7^5$ Checking with Alpha gives $7513995456 \gt 176473500$. The ratio is about $40$ instead of the $25$ I came up with in my estimate.