How many even positive integers are there that are divisors of 720? How many even positive integers are there that are divisors of 720 ?
I know how to compute the number of divisors but how to compute the number of even or odd positive divisors of a number ?
If we list the divisors of 720 (using mathematica) : {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45,48, 60, 72, 80, 90, 120, 144, 180, 240, 360, 720} among these only 24 are even,I am looking for some tricks that can be used in solving similar kinds of problems during exam (under a minute solution).

There is a very simple trick for this,first compute the prime factorization of $720$,which is $2^4 \times 3^2 \times 5$,the total number of factors here is $3 \times 2 \times 5 = 30$, and number of odd factors (number of factors of the odd primes)$=3 \times 2 = 6$,subtracting gives number of even factors = $24$.This method works for any number.
NOTE: If the number has no odd factors i.e,the prime factorization is of the form $2^a$,then the number of number of even factors is $a$ and number of odd factors is $1$.
