Number of divisors of $10800$ of form $4m+2$

Question

How many divisors of $10800$ are of form $4m+2$

MY TRY:

$10800=2^4.3^3.5^2$

Now any divisor of form $4m+2=2(2m+1)$

Now keeping in view the factorisation $m$ can be $1$ or $2$

Now How can I count the divisors???

Answer 1

You're nearly done.

Note that you are trying to to find $m$ such that $$4m+2|2^43^35^2$$ Now, erase two from both side, earning $$2m+1|2^33^35^2$$But since $m$ is odd, completely erase the $2$, getting us $$2m+1|3^35^2$$Thus there are $(3+1)(2+1)=12$ number of them.

Answer 2

If $d= 4m+2$ divides $10800$ then $2m+1$ divides $2^3 3^3 5^2$, and conversely. So you want to count the odd divisors of $2^3 3^3 5^2$, meaning the divisors of $3^3 5^2$.

This should be easy now.