Greatest Common Divisor of prime numbers

1,864

Solution 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 $2^4*3^3*5=2160$

Solution 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 $2^4 \times 3^3 \times 5$.

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Jack
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Updated on September 18, 2020

Comments

  • Jack
    Jack about 3 years

    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 $12^2 = 2^2\cdot2^2\cdot3^2$?

    • AnalyticHarmony
      AnalyticHarmony about 6 years
      Hint: $12^2 = (2^2*3)^2 = 2^4*3^2$
    • fleablood
      fleablood about 6 years
      that's a start. $12^2 =2^2*2^2*3^2=2^4*3^2$. And $15=3*5$ so $12^2*15=2^4*3^2*3*5 = 2^4*3^3*5$. Basically the first step is to rewrite everything as its unique prime factorization. $2^4*3^4*25*7 = 2^4*3^4*5^2*7$ and $2*12^2*15 = 2*2^4*3^2*3*5=2^4*3^3*5$. Then... you know what to do.
  • fleablood
    fleablood about 6 years
    Nicely explained and nice use of colors. The OP should note when you express a number and a product of prime powers, there will always be exactly one unique way to do it. So doing this will always be a failsafe that will work.