How does one find the GCD of three numbers with large exponent values?

Question

I have a problem which is to find the GCD of three numbers, each of which have large exponents:

$$ GCD( 2^{300}, 3^{200}, 2^{200})$$

What I have tried:

So far, I think there are two main steps here: (1) address the fact that its three numbers, and (2) attempt to get the GCD given the exponents. So first, I believe I can use Euclids algorithm:

Now, I would be working with only two values (which is what we have practiced so far). Now, I would have:

$$ GCD( 3^{200}, 2^{300})$$

What I am getting stuck is on the following step, and how to proceed, assuming this first step is correct. What I am inclined to do based on what we practice so far is to find the primes they have in common, but where I get stuck is how to do that for exponents, when searching for the GCD.

Any guidance would be appreciated!

how about $GCD( 3^{2}, 2^{3});$? - - - it is simpler, but not all that different. — 311411, Sep 22 '22 at 23:30
Hint. Does $3$ divide any power of $2$? Does $2$ divide any power of $3$? What is the gcd of $27$ and $64$? — Ethan Bolker, Sep 22 '22 at 23:30
Hint: What are the prime factorizations of these numbers? This should be easy. — Dan, Sep 22 '22 at 23:31
Also note that the gcd of $2^{200}$ and $2^{300}$ is $2^{200}$ (the number with the SMALLER exponent) , although this does not affect the final result. — Peter, Sep 23 '22 at 09:49

score 1 · Answer 1 · answered Sep 22 '22 at 23:31

1

If you have the prime factorisation of the numbers, there is a more efficient way to find the GCD than Euclid's algorithm - remember that if $d$ is a divisor of $a$ and $b$, that means that $a = md$ and $b = nd$ for some integers $m$ and $n$. But because of the Fundamental Theorem of Arithmetic, that means that the prime factors of $d$ must also be prime factors of $a$ and $b$, so it can only have prime factors that appear in both.

answered Sep 22 '22 at 23:31

ConMan

24,300

Please strive not to post more (dupe) answers to dupes of FAQs, cf. recent site policy announcement here. – Bill Dubuque Sep 22 '22 at 23:55

How does one find the GCD of three numbers with large exponent values?

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