List all the positive integer divisors of 3^s5^t, where s,t are integers and s,t 0. If r,s,t are positive integers, how many positive divisors does 2^r3^s5^t have? Please explain the answer, I am not sure how to get the result.

Question

List all the positive integer divisors of 3^s5^t, where s,t are integers and s,t >0. If r,s,t are positive integers, how many positive divisors does 2^r3^s5^t have? Please explain the answer, I am not sure how to get the result.

zepdrix · Answer

Oh boy.. what's a nice compact way to list all of these divisors.. hmm :d

zepdrix · Answer

take for example the number: $\large\rm 16875=3^3\cdot5^4$
This number is divisible by:
$\large\rm 3^1,3^2,3^3$
$\large\rm 5^1,5^2,5^3,5^4$
$\large\rm 3^1\cdot5^1,~3^1\cdot5^2,~3^1\cdot5^3,~3^1\cdot5^4$
$\large\rm 3^2\cdot5^1,~3^2\cdot5^2,~3^2\cdot5^3,~3^2\cdot5^4$
$\large\rm 3^3\cdot5^1,~3^3\cdot5^2,~3^3\cdot5^3$
You can see it gets kind of crazy ^
We need a nice way to write this.. hmm

zepdrix · Answer

Is this confusing? Maybe I can do a smaller number as an example:

zepdrix · Answer

$\large\rm 45=5\cdot9=5\cdot3\cdot3$
So this number is divisible by 3, 5, 3*3, 3*5, and 3*3*5.
So the divisors are 3,5,9,15, and 45.

zepdrix · Answer

Hmm sorry, I'm not sure :c

zepdrix · Answer

@ganeshie8 @Kainui @dan815  any ideas? 0_o

dan815 · Answer

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