Greatest Common Divisor of prime numbers

Question

What is the greatest common divisor of $2^4\cdot3^4\cdot25\cdot7$ and $2\cdot12^2\cdot15$?

I know how to find the GCD of a problem like this, but I didn't know what to do since $12$ is not a prime number. Should I say $12^2 = 2^2\cdot2^2\cdot3^2$?

Answer 1

The first one equal to $2^4*3^4*5^2*7$

The second one equals to $2^5*3^3*5$

Therefore the GCD is $2^4*3^3*5=2160$

Answer 2

Yes, that is precisely what you should be doing.

In short, when you want to find the gcd of two numbers directly, you must first represent each one as a product of prime powers, before matching powers and primes.

So in our case, we can write: $$ 2^4 \times 3^4 \times \color{blue}{25} \times 7 = 2^4 \times 3^4 \times \color{blue}{5^2} \times 7 \\ 2 \times \color{red}{12^2} \times \color{green}{15} = 2 \times \color{red}{2^2 \times 2^2 \times 3^2} \times \color{green}{3 \times 5} = 2^5 \times 3^3 \times 5 $$

where, I highlight by color the terms that are expanded on both the LHS and RHS.

Now, we are permitted to compare powers directly. This gives us the answer as $2^4 \times 3^3 \times 5$.