What is the greatest common divisor of 300 middot 35 middot

What is the greatest common divisor of 300 middot 35 middot 7^5 and 33^5 middot 3 middot 64? Obtain a single number.

Solution

Factoring the given numbers we get,

300*35*7^5=(2^2×3×5^2)*(7*5)*7^5=2^2*3*5^3*7^6

33^5*3*64=(3*11)^5*3*2^6=3^5*11^5*3*2^6=2^6*3^6*11^5

Now, we see that 2^2 and 3 are present in both the numbers that means they divide both the given numbers and, hence the greatest common divisor if 2^2*3=4*3=12